38:
941:
924:
1584:
3049:
3550:
2787:
2932:
2513:
2205:
2346:
2659:
3187:
1929:
1464:
3321:
For an equation to be meaningful, the coefficient of at least one variable must be non-zero. If every variable has a zero coefficient, then, as mentioned for one variable, the equation is either
2087:
1469:
784:
136:
242:
2940:
1703:
304:
182:
1837:
1770:
1180:
1116:
3419:
557:
871:
832:
1342:
1396:
1636:
1272:
358:
1453:
620:
1014:
711:
2667:
518:
492:
3289:
383:
in the
Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables. This is the origin of the term
3262:
3309:
3797:
3733:
3711:
2795:
2376:
2095:
1850:
A line that is not parallel to an axis and does not pass through the origin cuts the axes into two different points. The intercept values
889:
is a function that maps a sum to the sum of the images of the summands. So, for this definition, the above function is linear only when
896:, that is when the line passes through the origin. To avoid confusion, the functions whose graph is an arbitrary line are often called
3977:
3563:
2220:
3318:-tuple such that substituting each element of the tuple for the corresponding variable transforms the equation into a true equality.
2536:
3091:
1579:{\displaystyle {\begin{aligned}m&=-{\frac {a}{b}},\\x_{0}&=-{\frac {c}{a}},\\y_{0}&=-{\frac {c}{b}}.\end{aligned}}}
3054:
Besides being very simple and mnemonic, this form has the advantage of being a special case of the more general equation of a
1874:
1189:
There are various ways of defining a line. In the following subsections, a linear equation of the line is given in each case.
2370:
This form is not symmetric in the two given points, but a symmetric form can be obtained by regrouping the constant terms:
3768:
2018:
3972:
3956:
734:
60:
31:
4028:
3992:
3763:
3579:
3383:
if the coefficients are complex numbers or belong to any field). In the case of three variables, this hyperplane is a
1996:, there is exactly one line that passes through them. There are several ways to write a linear equation of this line.
3837:
654:
441:
3790:
3982:
257:
3921:
3044:{\displaystyle {\begin{vmatrix}x&y&1\\x_{1}&y_{1}&1\\x_{2}&y_{2}&1\end{vmatrix}}=0.}
195:
3951:
3946:
3941:
3819:
3545:{\displaystyle x_{j}=-{\frac {b}{a_{j}}}-\sum _{i\in \{1,\ldots ,n\},i\neq j}{\frac {a_{i}}{a_{j}}}x_{i}.}
790:
185:
1644:
263:
141:
3931:
3911:
3354:
2211:
1781:
1714:
1146:
1082:
1020:
372:
189:
526:
843:
260:, provided they do not contain any of the variables. To yield a meaningful equation, the coefficients
3997:
3916:
3783:
3758:
3560:
807:
1042:
The phrase "linear equation" takes its origin in this correspondence between lines and equations: a
324:
of such an equation are the values that, when substituted for the unknowns, make the equality true.
3810:
1402:
1296:
1057:
802:
794:
437:
321:
314:
1353:
4033:
3854:
3849:
3584:
2782:{\displaystyle {\begin{vmatrix}x-x_{1}&y-y_{1}\\x_{2}-x_{1}&y_{2}-y_{1}\end{vmatrix}}=0.}
1842:
for emphasizing that the slope of a line can be computed from the coordinates of any two points.
1601:
1232:
330:
3987:
1411:
578:
975:
672:
3832:
3729:
3707:
3686:
3594:
3070:
422:
310:
3666:
497:
459:
4007:
3885:
3878:
3873:
3599:
3589:
3384:
3074:
2367:(for verifying this, it suffices to verify that the two given points satisfy the equation).
1039:
are not both zero. Conversely, every line is the set of all solutions of a linear equation.
387:
for describing this type of equation. More generally, the solutions of a linear equation in
3267:
2934:
can be obtained by expanding with respect to its first row the determinant in the equation
3895:
3890:
3827:
3376:
1028:
1024:
898:
886:
798:
436:
solutions and, more generally, to linear equations with coefficients and solutions in any
403:
380:
376:
3241:
428:
This article considers the case of a single equation with coefficients from the field of
364:
refers implicitly to this particular case, in which the variable is sensibly called the
3866:
3861:
3775:
3294:
882:
433:
3085:
A linear equation with more than two variables may always be assumed to have the form
37:
4022:
3722:
2518:(exchanging the two points changes the sign of the left-hand side of the equation).
4002:
3380:
2526:
The two-point form of the equation of a line can be expressed simply in terms of a
3704:
College
Mathematics for Business, Economics, Life Sciences and the Social Sciences
3679:
3659:
1277:
If, moreover, the line is not horizontal, it can be defined by its slope and its
3556:
2527:
835:
429:
418:
249:
245:
46:
17:
413:
Linear equations occur frequently in all mathematics and their applications in
327:
In the case of just one variable, there is exactly one solution (provided that
3806:
3366:
3055:
911:
392:
3936:
253:
432:, for which one studies the real solutions. All of its content applies to
3926:
1638:
of any point of the line. In this case, a linear equation of the line is
878:
54:
2927:{\displaystyle (y_{1}-y_{2})x+(x_{2}-x_{1})y+(x_{1}y_{2}-x_{2}y_{1})=0}
2508:{\displaystyle (y_{1}-y_{2})x+(x_{2}-x_{1})y+(x_{1}y_{2}-x_{2}y_{1})=0}
1401:
These forms rely on the habit of considering a nonvertical line as the
414:
309:
Alternatively, a linear equation can be obtained by equating to zero a
371:
In the case of two variables, each solution may be interpreted as the
2200:{\displaystyle y-y_{1}={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}(x-x_{1}).}
940:
939:
923:
922:
1934:(It is easy to verify that the line defined by this equation has
1027:. With this interpretation, all solutions of the equation form a
2341:{\displaystyle (x_{2}-x_{1})(y-y_{1})-(y_{2}-y_{1})(x-x_{1})=0,}
1868:
of these two points are nonzero, and an equation of the line is
3779:
2654:{\displaystyle (x_{2}-x_{1})(y-y_{1})-(y_{2}-y_{1})(x-x_{1})=0}
27:
Equation that does not involve powers or products of variables
3182:{\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}+b=0.}
1046:
in two variables is an equation whose solutions form a line.
440:. For the case of several simultaneous linear equations, see
2661:
is the result of expanding the determinant in the equation
873:
The functions whose graph is a line are generally called
1924:{\displaystyle {\frac {x}{x_{0}}}+{\frac {y}{y_{0}}}=1.}
2949:
2676:
1458:
these forms can be easily deduced from the relations
3422:
3297:
3270:
3244:
3094:
2943:
2798:
2670:
2539:
2379:
2223:
2098:
2021:
1877:
1784:
1717:
1647:
1604:
1467:
1414:
1356:
1299:
1235:
1149:
1085:
978:
846:
810:
737:
675:
581:
529:
500:
462:
333:
266:
198:
144:
63:
3706:(11th ed.), Upper Saddle River, N.J.: Pearson,
3618:
645:
are real numbers, it has infinitely many solutions.
3965:
3904:
3817:
3702:Barnett, R.A.; Ziegler, M.R.; Byleen, K.E. (2008),
3346:-tuples that are solutions of a linear equation in
2082:{\displaystyle {\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.}
1064:that has been defined in the preceding section. If
3721:
3544:
3303:
3283:
3256:
3213:in old books). Depending on the context, the term
3181:
3043:
2926:
2781:
2653:
2507:
2340:
2199:
2081:
1923:
1831:
1764:
1697:
1630:
1578:
1447:
1390:
1336:
1266:
1174:
1110:
1008:
865:
826:
779:{\displaystyle y=-{\frac {a}{b}}x-{\frac {c}{b}}.}
778:
705:
614:
551:
512:
486:
352:
298:
236:
176:
131:{\displaystyle a_{1}x_{1}+\ldots +a_{n}x_{n}+b=0,}
130:
425:are often well approximated by linear equations.
1594:A non-vertical line can be defined by its slope
1197:A non-vertical line can be defined by its slope
1193:Slope–intercept form or Gradient-intercept form
3630:
41:Two graphs of linear equations in two variables
3828:Zero polynomial (degree undefined or −1 or −∞)
3641:
3639:
3791:
8:
3645:
3488:
3470:
1290:. In this case, its equation can be written
716:is a linear equation in the single variable
379:. The solutions of a linear equation form a
3661:An Elementary Course in Theory of Equations
3069:. These equations rely on the condition of
3798:
3784:
3776:
3681:Numbers Universalized: An Advanced Algebra
3277:
3533:
3521:
3511:
3505:
3463:
3448:
3439:
3427:
3421:
3296:
3269:
3243:
3161:
3151:
3132:
3122:
3109:
3099:
3093:
3016:
3004:
2985:
2973:
2944:
2942:
2909:
2899:
2886:
2876:
2854:
2841:
2819:
2806:
2797:
2759:
2746:
2734:
2721:
2707:
2689:
2671:
2669:
2636:
2614:
2601:
2582:
2560:
2547:
2538:
2490:
2480:
2467:
2457:
2435:
2422:
2400:
2387:
2378:
2320:
2298:
2285:
2266:
2244:
2231:
2222:
2185:
2163:
2150:
2138:
2125:
2118:
2109:
2097:
2067:
2054:
2042:
2029:
2022:
2020:
1907:
1898:
1887:
1878:
1876:
1820:
1795:
1783:
1753:
1737:
1716:
1683:
1658:
1646:
1622:
1609:
1603:
1559:
1543:
1522:
1506:
1485:
1468:
1466:
1413:
1379:
1355:
1322:
1298:
1255:
1234:
1159:
1148:
1132:, the line is the graph of a function of
1095:
1084:
977:
850:
845:
814:
809:
763:
747:
736:
724:. It has therefore a unique solution for
674:
580:
539:
528:
499:
461:
338:
332:
317:, from which the coefficients are taken.
290:
271:
265:
228:
209:
197:
168:
149:
143:
107:
97:
78:
68:
62:
1218:coordinate of its intersection with the
1143:, one has a horizontal line of equation
1118:which is not the graph of a function of
252:. The coefficients may be considered as
36:
3720:Larson, Ron; Hostetler, Robert (2007),
3611:
1590:Point–slope form or Point-gradient form
3402:, then the equation can be solved for
2530:. There are two common ways for that.
3684:. American Book Company. p. 113.
902:, and the linear functions such that
256:of the equation and may be arbitrary
237:{\displaystyle b,a_{1},\ldots ,a_{n}}
7:
3314:A solution of such an equation is a
3742:Wilson, W.A.; Tracey, J.I. (1925),
3390:If a linear equation is given with
1698:{\displaystyle y=y_{1}+m(x-x_{1}),}
1405:. For a line given by an equation
567:A linear equation in two variables
299:{\displaystyle a_{1},\ldots ,a_{n}}
177:{\displaystyle x_{1},\ldots ,x_{n}}
3664:. J. Wiley & sons. p. 17.
3619:Barnett, Ziegler & Byleen 2008
1832:{\displaystyle y-y_{1}=m(x-x_{1})}
1775:This equation can also be written
1765:{\displaystyle y=mx+y_{1}-mx_{1}.}
1175:{\displaystyle y=-{\frac {c}{b}}.}
1111:{\displaystyle x=-{\frac {c}{a}},}
452:A linear equation in one variable
25:
552:{\displaystyle x=-{\frac {b}{a}}}
306:are required to not all be zero.
3332:) as having no solution, or all
1075:(that is a line parallel to the
866:{\displaystyle -{\frac {c}{b}}.}
3264:variables, it is common to use
3062:points in a space of dimension
827:{\displaystyle -{\frac {a}{b}}}
3746:(revised ed.), D.C. Heath
3678:David Martin Sensenig (1890).
3658:Charles Hiram Chapman (1892).
3311:instead of indexed variables.
2915:
2869:
2860:
2834:
2825:
2799:
2642:
2623:
2620:
2594:
2588:
2569:
2566:
2540:
2496:
2450:
2441:
2415:
2406:
2380:
2326:
2307:
2304:
2278:
2272:
2253:
2250:
2224:
2191:
2172:
1826:
1807:
1689:
1670:
1328:
1309:
1:
3988:Horner's method of evaluation
1337:{\displaystyle y=m(x-x_{0}),}
3724:Precalculus:A Concise Course
2089:Thus, a point-slope form is
1391:{\displaystyle y=mx-mx_{0}.}
944:Horizontal line of equation
57:that may be put in the form
32:Linear differential equation
3993:Polynomial identity testing
3764:Encyclopedia of Mathematics
3631:Larson & Hostetler 2007
3580:Linear equation over a ring
2015:, the slope of the line is
1960:Given two different points
1631:{\displaystyle x_{1},y_{1}}
1267:{\displaystyle y=mx+y_{0}.}
353:{\displaystyle a_{1}\neq 0}
4050:
1448:{\displaystyle ax+by+c=0,}
1222:-axis). In this case, its
927:Vertical line of equation
655:Linear function (calculus)
652:
615:{\displaystyle ax+by+c=0,}
442:system of linear equations
29:
2351:which is valid also when
2214:, one gets the equation
1009:{\displaystyle ax+by+c=0}
706:{\displaystyle ax+by+c=0}
395:(a subspace of dimension
3646:Wilson & Tracey 1925
3555:If the coefficients are
3217:can be reserved for the
919:Geometric interpretation
30:Not to be confused with
3978:Greatest common divisor
3081:More than two variables
513:{\displaystyle a\neq 0}
487:{\displaystyle ax+b=0,}
3850:Quadratic function (2)
3546:
3305:
3285:
3258:
3183:
3045:
2928:
2783:
2655:
2509:
2342:
2201:
2083:
1952:as intercept values).
1925:
1833:
1766:
1699:
1632:
1598:, and the coordinates
1580:
1449:
1392:
1338:
1268:
1176:
1112:
1010:
954:
937:
867:
828:
797:of this function is a
780:
707:
616:
553:
514:
488:
354:
300:
238:
178:
132:
42:
3833:Constant function (0)
3547:
3355:Cartesian coordinates
3306:
3286:
3284:{\displaystyle x,\;y}
3259:
3184:
3046:
2929:
2784:
2656:
2510:
2343:
2212:clearing denominators
2202:
2084:
1926:
1834:
1767:
1700:
1633:
1581:
1450:
1393:
1339:
1269:
1177:
1113:
1058:graph of the function
1021:Cartesian coordinates
1019:may be viewed as the
1011:
969:of a linear equation
943:
926:
868:
829:
781:
728:, which is given by
708:
617:
554:
515:
489:
373:Cartesian coordinates
355:
301:
239:
179:
133:
40:
3966:Tools and algorithms
3886:Quintic function (5)
3874:Quartic function (4)
3811:polynomial functions
3728:, Houghton Mifflin,
3420:
3357:of the points of an
3295:
3268:
3242:
3092:
2941:
2796:
2668:
2537:
2377:
2221:
2096:
2019:
1875:
1782:
1715:
1645:
1602:
1465:
1412:
1354:
1297:
1233:
1147:
1083:
976:
844:
808:
735:
673:
579:
527:
498:
460:
331:
264:
196:
142:
61:
3896:Septic equation (7)
3891:Sextic equation (6)
3838:Linear function (1)
3687:Extract of page 113
3257:{\displaystyle n=3}
1403:graph of a function
1079:-axis) of equation
720:for every value of
360:). Often, the term
4029:Elementary algebra
3862:Cubic function (3)
3855:Quadratic equation
3667:Extract of page 17
3600:Nonlinear equation
3585:Algebraic equation
3542:
3504:
3301:
3281:
3254:
3238:When dealing with
3179:
3041:
3029:
2924:
2779:
2767:
2651:
2505:
2338:
2197:
2079:
1921:
1829:
1762:
1695:
1628:
1576:
1574:
1445:
1388:
1347:or, equivalently,
1334:
1264:
1185:Equation of a line
1172:
1108:
1056:, the line is the
1023:of a point in the
1006:
955:
938:
877:in the context of
863:
824:
776:
703:
612:
575:can be written as
549:
510:
484:
456:can be written as
423:non-linear systems
375:of a point of the
350:
296:
248:, which are often
234:
174:
128:
43:
4016:
4015:
3957:Quasi-homogeneous
3759:"Linear equation"
3744:Analytic Geometry
3735:978-0-618-62719-6
3713:978-0-13-157225-6
3595:Linear inequality
3559:, this defines a
3527:
3459:
3454:
3304:{\displaystyle z}
3071:linear dependence
2170:
2074:
1913:
1893:
1567:
1530:
1493:
1167:
1103:
909:are often called
858:
822:
771:
755:
547:
421:, partly because
391:variables form a
311:linear polynomial
16:(Redirected from
4041:
3879:Quartic equation
3800:
3793:
3786:
3777:
3772:
3747:
3738:
3727:
3716:
3689:
3685:
3675:
3669:
3665:
3655:
3649:
3643:
3634:
3628:
3622:
3616:
3590:Line coordinates
3567:
3551:
3549:
3548:
3543:
3538:
3537:
3528:
3526:
3525:
3516:
3515:
3506:
3503:
3455:
3453:
3452:
3440:
3432:
3431:
3412:
3401:
3375:
3373:
3364:
3352:
3350:
3345:
3338:
3336:
3331:
3317:
3310:
3308:
3307:
3302:
3290:
3288:
3287:
3282:
3263:
3261:
3260:
3255:
3234:
3227:
3204:
3196:, often denoted
3195:
3192:The coefficient
3188:
3186:
3185:
3180:
3166:
3165:
3156:
3155:
3137:
3136:
3127:
3126:
3114:
3113:
3104:
3103:
3075:projective space
3068:
3061:
3058:passing through
3050:
3048:
3047:
3042:
3034:
3033:
3021:
3020:
3009:
3008:
2990:
2989:
2978:
2977:
2933:
2931:
2930:
2925:
2914:
2913:
2904:
2903:
2891:
2890:
2881:
2880:
2859:
2858:
2846:
2845:
2824:
2823:
2811:
2810:
2788:
2786:
2785:
2780:
2772:
2771:
2764:
2763:
2751:
2750:
2739:
2738:
2726:
2725:
2712:
2711:
2694:
2693:
2660:
2658:
2657:
2652:
2641:
2640:
2619:
2618:
2606:
2605:
2587:
2586:
2565:
2564:
2552:
2551:
2522:Determinant form
2514:
2512:
2511:
2506:
2495:
2494:
2485:
2484:
2472:
2471:
2462:
2461:
2440:
2439:
2427:
2426:
2405:
2404:
2392:
2391:
2366:
2347:
2345:
2344:
2339:
2325:
2324:
2303:
2302:
2290:
2289:
2271:
2270:
2249:
2248:
2236:
2235:
2206:
2204:
2203:
2198:
2190:
2189:
2171:
2169:
2168:
2167:
2155:
2154:
2144:
2143:
2142:
2130:
2129:
2119:
2114:
2113:
2088:
2086:
2085:
2080:
2075:
2073:
2072:
2071:
2059:
2058:
2048:
2047:
2046:
2034:
2033:
2023:
2014:
1995:
1977:
1951:
1942:
1930:
1928:
1927:
1922:
1914:
1912:
1911:
1899:
1894:
1892:
1891:
1879:
1867:
1858:
1838:
1836:
1835:
1830:
1825:
1824:
1800:
1799:
1771:
1769:
1768:
1763:
1758:
1757:
1742:
1741:
1704:
1702:
1701:
1696:
1688:
1687:
1663:
1662:
1637:
1635:
1634:
1629:
1627:
1626:
1614:
1613:
1597:
1585:
1583:
1582:
1577:
1575:
1568:
1560:
1548:
1547:
1531:
1523:
1511:
1510:
1494:
1486:
1454:
1452:
1451:
1446:
1397:
1395:
1394:
1389:
1384:
1383:
1343:
1341:
1340:
1335:
1327:
1326:
1289:
1280:
1273:
1271:
1270:
1265:
1260:
1259:
1226:can be written
1221:
1217:
1213:
1204:
1200:
1181:
1179:
1178:
1173:
1168:
1160:
1142:
1135:
1131:
1121:
1117:
1115:
1114:
1109:
1104:
1096:
1078:
1071:, the line is a
1070:
1063:
1055:
1038:
1034:
1031:, provided that
1015:
1013:
1012:
1007:
968:
953:
936:
908:
899:affine functions
895:
875:linear functions
872:
870:
869:
864:
859:
851:
838:
833:
831:
830:
825:
823:
815:
785:
783:
782:
777:
772:
764:
756:
748:
727:
723:
719:
712:
710:
709:
704:
666:, the equation
665:
644:
640:
633:
629:
625:
621:
619:
618:
613:
574:
570:
558:
556:
555:
550:
548:
540:
523:The solution is
519:
517:
516:
511:
493:
491:
490:
485:
455:
409:
401:
390:
359:
357:
356:
351:
343:
342:
305:
303:
302:
297:
295:
294:
276:
275:
243:
241:
240:
235:
233:
232:
214:
213:
183:
181:
180:
175:
173:
172:
154:
153:
137:
135:
134:
129:
112:
111:
102:
101:
83:
82:
73:
72:
21:
18:Linear equations
4049:
4048:
4044:
4043:
4042:
4040:
4039:
4038:
4019:
4018:
4017:
4012:
3961:
3900:
3843:Linear equation
3813:
3804:
3757:
3754:
3741:
3736:
3719:
3714:
3701:
3698:
3693:
3692:
3677:
3676:
3672:
3657:
3656:
3652:
3644:
3637:
3629:
3625:
3617:
3613:
3608:
3576:
3565:
3529:
3517:
3507:
3444:
3423:
3418:
3417:
3411:
3403:
3399:
3391:
3377:Euclidean space
3371:
3370:
3358:
3348:
3347:
3343:
3339:are solutions.
3334:
3333:
3326:
3315:
3293:
3292:
3266:
3265:
3240:
3239:
3229:
3226:
3218:
3209:(sometimes the
3203:
3197:
3193:
3157:
3147:
3128:
3118:
3105:
3095:
3090:
3089:
3083:
3073:of points in a
3063:
3059:
3028:
3027:
3022:
3012:
3010:
3000:
2997:
2996:
2991:
2981:
2979:
2969:
2966:
2965:
2960:
2955:
2945:
2939:
2938:
2905:
2895:
2882:
2872:
2850:
2837:
2815:
2802:
2794:
2793:
2766:
2765:
2755:
2742:
2740:
2730:
2717:
2714:
2713:
2703:
2695:
2685:
2672:
2666:
2665:
2632:
2610:
2597:
2578:
2556:
2543:
2535:
2534:
2524:
2486:
2476:
2463:
2453:
2431:
2418:
2396:
2383:
2375:
2374:
2365:
2358:
2352:
2316:
2294:
2281:
2262:
2240:
2227:
2219:
2218:
2181:
2159:
2146:
2145:
2134:
2121:
2120:
2105:
2094:
2093:
2063:
2050:
2049:
2038:
2025:
2024:
2017:
2016:
2013:
2006:
2000:
1993:
1986:
1979:
1975:
1968:
1961:
1958:
1950:
1944:
1941:
1935:
1903:
1883:
1873:
1872:
1866:
1860:
1857:
1851:
1848:
1816:
1791:
1780:
1779:
1749:
1733:
1713:
1712:
1679:
1654:
1643:
1642:
1618:
1605:
1600:
1599:
1595:
1592:
1573:
1572:
1549:
1539:
1536:
1535:
1512:
1502:
1499:
1498:
1475:
1463:
1462:
1410:
1409:
1375:
1352:
1351:
1318:
1295:
1294:
1288:
1282:
1278:
1251:
1231:
1230:
1224:linear equation
1219:
1215:
1212:
1206:
1202:
1198:
1195:
1187:
1145:
1144:
1137:
1133:
1126:
1119:
1081:
1080:
1076:
1065:
1061:
1050:
1044:linear equation
1036:
1032:
1025:Euclidean plane
974:
973:
958:
945:
928:
921:
903:
890:
887:linear function
842:
841:
836:
806:
805:
789:This defines a
733:
732:
725:
721:
717:
671:
670:
660:
657:
651:
649:Linear function
642:
638:
631:
627:
623:
577:
576:
572:
568:
565:
525:
524:
496:
495:
458:
457:
453:
450:
407:
404:Euclidean space
396:
388:
377:Euclidean plane
362:linear equation
334:
329:
328:
286:
267:
262:
261:
224:
205:
194:
193:
164:
145:
140:
139:
103:
93:
74:
64:
59:
58:
51:linear equation
35:
28:
23:
22:
15:
12:
11:
5:
4047:
4045:
4037:
4036:
4031:
4021:
4020:
4014:
4013:
4011:
4010:
4005:
4000:
3995:
3990:
3985:
3980:
3975:
3969:
3967:
3963:
3962:
3960:
3959:
3954:
3949:
3944:
3939:
3934:
3929:
3924:
3919:
3914:
3908:
3906:
3902:
3901:
3899:
3898:
3893:
3888:
3883:
3882:
3881:
3871:
3870:
3869:
3867:Cubic equation
3859:
3858:
3857:
3847:
3846:
3845:
3835:
3830:
3824:
3822:
3815:
3814:
3805:
3803:
3802:
3795:
3788:
3780:
3774:
3773:
3753:
3752:External links
3750:
3749:
3748:
3739:
3734:
3717:
3712:
3697:
3694:
3691:
3690:
3670:
3650:
3635:
3623:
3610:
3609:
3607:
3604:
3603:
3602:
3597:
3592:
3587:
3582:
3575:
3572:
3568:real variables
3553:
3552:
3541:
3536:
3532:
3524:
3520:
3514:
3510:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3466:
3462:
3458:
3451:
3447:
3443:
3438:
3435:
3430:
3426:
3407:
3395:
3300:
3280:
3276:
3273:
3253:
3250:
3247:
3222:
3205:is called the
3201:
3190:
3189:
3178:
3175:
3172:
3169:
3164:
3160:
3154:
3150:
3146:
3143:
3140:
3135:
3131:
3125:
3121:
3117:
3112:
3108:
3102:
3098:
3082:
3079:
3052:
3051:
3040:
3037:
3032:
3026:
3023:
3019:
3015:
3011:
3007:
3003:
2999:
2998:
2995:
2992:
2988:
2984:
2980:
2976:
2972:
2968:
2967:
2964:
2961:
2959:
2956:
2954:
2951:
2950:
2948:
2923:
2920:
2917:
2912:
2908:
2902:
2898:
2894:
2889:
2885:
2879:
2875:
2871:
2868:
2865:
2862:
2857:
2853:
2849:
2844:
2840:
2836:
2833:
2830:
2827:
2822:
2818:
2814:
2809:
2805:
2801:
2790:
2789:
2778:
2775:
2770:
2762:
2758:
2754:
2749:
2745:
2741:
2737:
2733:
2729:
2724:
2720:
2716:
2715:
2710:
2706:
2702:
2699:
2696:
2692:
2688:
2684:
2681:
2678:
2677:
2675:
2650:
2647:
2644:
2639:
2635:
2631:
2628:
2625:
2622:
2617:
2613:
2609:
2604:
2600:
2596:
2593:
2590:
2585:
2581:
2577:
2574:
2571:
2568:
2563:
2559:
2555:
2550:
2546:
2542:
2523:
2520:
2516:
2515:
2504:
2501:
2498:
2493:
2489:
2483:
2479:
2475:
2470:
2466:
2460:
2456:
2452:
2449:
2446:
2443:
2438:
2434:
2430:
2425:
2421:
2417:
2414:
2411:
2408:
2403:
2399:
2395:
2390:
2386:
2382:
2363:
2356:
2349:
2348:
2337:
2334:
2331:
2328:
2323:
2319:
2315:
2312:
2309:
2306:
2301:
2297:
2293:
2288:
2284:
2280:
2277:
2274:
2269:
2265:
2261:
2258:
2255:
2252:
2247:
2243:
2239:
2234:
2230:
2226:
2208:
2207:
2196:
2193:
2188:
2184:
2180:
2177:
2174:
2166:
2162:
2158:
2153:
2149:
2141:
2137:
2133:
2128:
2124:
2117:
2112:
2108:
2104:
2101:
2078:
2070:
2066:
2062:
2057:
2053:
2045:
2041:
2037:
2032:
2028:
2011:
2004:
1991:
1984:
1973:
1966:
1957:
1956:Two-point form
1954:
1948:
1939:
1932:
1931:
1920:
1917:
1910:
1906:
1902:
1897:
1890:
1886:
1882:
1864:
1855:
1847:
1846:Intercept form
1844:
1840:
1839:
1828:
1823:
1819:
1815:
1812:
1809:
1806:
1803:
1798:
1794:
1790:
1787:
1773:
1772:
1761:
1756:
1752:
1748:
1745:
1740:
1736:
1732:
1729:
1726:
1723:
1720:
1706:
1705:
1694:
1691:
1686:
1682:
1678:
1675:
1672:
1669:
1666:
1661:
1657:
1653:
1650:
1625:
1621:
1617:
1612:
1608:
1591:
1588:
1587:
1586:
1571:
1566:
1563:
1558:
1555:
1552:
1550:
1546:
1542:
1538:
1537:
1534:
1529:
1526:
1521:
1518:
1515:
1513:
1509:
1505:
1501:
1500:
1497:
1492:
1489:
1484:
1481:
1478:
1476:
1474:
1471:
1470:
1456:
1455:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1399:
1398:
1387:
1382:
1378:
1374:
1371:
1368:
1365:
1362:
1359:
1345:
1344:
1333:
1330:
1325:
1321:
1317:
1314:
1311:
1308:
1305:
1302:
1286:
1275:
1274:
1263:
1258:
1254:
1250:
1247:
1244:
1241:
1238:
1210:
1194:
1191:
1186:
1183:
1171:
1166:
1163:
1158:
1155:
1152:
1125:Similarly, if
1107:
1102:
1099:
1094:
1091:
1088:
1017:
1016:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
957:Each solution
920:
917:
883:linear algebra
881:. However, in
862:
857:
854:
849:
821:
818:
813:
787:
786:
775:
770:
767:
762:
759:
754:
751:
746:
743:
740:
714:
713:
702:
699:
696:
693:
690:
687:
684:
681:
678:
653:Main article:
650:
647:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
564:
561:
546:
543:
538:
535:
532:
509:
506:
503:
483:
480:
477:
474:
471:
468:
465:
449:
446:
349:
346:
341:
337:
293:
289:
285:
282:
279:
274:
270:
231:
227:
223:
220:
217:
212:
208:
204:
201:
171:
167:
163:
160:
157:
152:
148:
127:
124:
121:
118:
115:
110:
106:
100:
96:
92:
89:
86:
81:
77:
71:
67:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4046:
4035:
4032:
4030:
4027:
4026:
4024:
4009:
4008:Gröbner basis
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3974:
3973:Factorization
3971:
3970:
3968:
3964:
3958:
3955:
3953:
3950:
3948:
3945:
3943:
3940:
3938:
3935:
3933:
3930:
3928:
3925:
3923:
3920:
3918:
3915:
3913:
3910:
3909:
3907:
3905:By properties
3903:
3897:
3894:
3892:
3889:
3887:
3884:
3880:
3877:
3876:
3875:
3872:
3868:
3865:
3864:
3863:
3860:
3856:
3853:
3852:
3851:
3848:
3844:
3841:
3840:
3839:
3836:
3834:
3831:
3829:
3826:
3825:
3823:
3821:
3816:
3812:
3808:
3801:
3796:
3794:
3789:
3787:
3782:
3781:
3778:
3770:
3766:
3765:
3760:
3756:
3755:
3751:
3745:
3740:
3737:
3731:
3726:
3725:
3718:
3715:
3709:
3705:
3700:
3699:
3695:
3688:
3683:
3682:
3674:
3671:
3668:
3663:
3662:
3654:
3651:
3647:
3642:
3640:
3636:
3632:
3627:
3624:
3620:
3615:
3612:
3605:
3601:
3598:
3596:
3593:
3591:
3588:
3586:
3583:
3581:
3578:
3577:
3573:
3571:
3569:
3562:
3558:
3539:
3534:
3530:
3522:
3518:
3512:
3508:
3500:
3497:
3494:
3491:
3485:
3482:
3479:
3476:
3473:
3467:
3464:
3460:
3456:
3449:
3445:
3441:
3436:
3433:
3428:
3424:
3416:
3415:
3414:
3410:
3406:
3398:
3394:
3388:
3386:
3382:
3378:
3368:
3365:-dimensional
3362:
3356:
3340:
3329:
3324:
3319:
3312:
3298:
3278:
3274:
3271:
3251:
3248:
3245:
3236:
3232:
3225:
3221:
3216:
3212:
3211:absolute term
3208:
3207:constant term
3200:
3176:
3173:
3170:
3167:
3162:
3158:
3152:
3148:
3144:
3141:
3138:
3133:
3129:
3123:
3119:
3115:
3110:
3106:
3100:
3096:
3088:
3087:
3086:
3080:
3078:
3076:
3072:
3066:
3057:
3038:
3035:
3030:
3024:
3017:
3013:
3005:
3001:
2993:
2986:
2982:
2974:
2970:
2962:
2957:
2952:
2946:
2937:
2936:
2935:
2921:
2918:
2910:
2906:
2900:
2896:
2892:
2887:
2883:
2877:
2873:
2866:
2863:
2855:
2851:
2847:
2842:
2838:
2831:
2828:
2820:
2816:
2812:
2807:
2803:
2792:The equation
2776:
2773:
2768:
2760:
2756:
2752:
2747:
2743:
2735:
2731:
2727:
2722:
2718:
2708:
2704:
2700:
2697:
2690:
2686:
2682:
2679:
2673:
2664:
2663:
2662:
2648:
2645:
2637:
2633:
2629:
2626:
2615:
2611:
2607:
2602:
2598:
2591:
2583:
2579:
2575:
2572:
2561:
2557:
2553:
2548:
2544:
2533:The equation
2531:
2529:
2521:
2519:
2502:
2499:
2491:
2487:
2481:
2477:
2473:
2468:
2464:
2458:
2454:
2447:
2444:
2436:
2432:
2428:
2423:
2419:
2412:
2409:
2401:
2397:
2393:
2388:
2384:
2373:
2372:
2371:
2368:
2362:
2355:
2335:
2332:
2329:
2321:
2317:
2313:
2310:
2299:
2295:
2291:
2286:
2282:
2275:
2267:
2263:
2259:
2256:
2245:
2241:
2237:
2232:
2228:
2217:
2216:
2215:
2213:
2194:
2186:
2182:
2178:
2175:
2164:
2160:
2156:
2151:
2147:
2139:
2135:
2131:
2126:
2122:
2115:
2110:
2106:
2102:
2099:
2092:
2091:
2090:
2076:
2068:
2064:
2060:
2055:
2051:
2043:
2039:
2035:
2030:
2026:
2010:
2003:
1997:
1990:
1983:
1972:
1965:
1955:
1953:
1947:
1938:
1918:
1915:
1908:
1904:
1900:
1895:
1888:
1884:
1880:
1871:
1870:
1869:
1863:
1854:
1845:
1843:
1821:
1817:
1813:
1810:
1804:
1801:
1796:
1792:
1788:
1785:
1778:
1777:
1776:
1759:
1754:
1750:
1746:
1743:
1738:
1734:
1730:
1727:
1724:
1721:
1718:
1711:
1710:
1709:
1692:
1684:
1680:
1676:
1673:
1667:
1664:
1659:
1655:
1651:
1648:
1641:
1640:
1639:
1623:
1619:
1615:
1610:
1606:
1589:
1569:
1564:
1561:
1556:
1553:
1551:
1544:
1540:
1532:
1527:
1524:
1519:
1516:
1514:
1507:
1503:
1495:
1490:
1487:
1482:
1479:
1477:
1472:
1461:
1460:
1459:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1408:
1407:
1406:
1404:
1385:
1380:
1376:
1372:
1369:
1366:
1363:
1360:
1357:
1350:
1349:
1348:
1331:
1323:
1319:
1315:
1312:
1306:
1303:
1300:
1293:
1292:
1291:
1285:
1261:
1256:
1252:
1248:
1245:
1242:
1239:
1236:
1229:
1228:
1227:
1225:
1209:
1192:
1190:
1184:
1182:
1169:
1164:
1161:
1156:
1153:
1150:
1140:
1129:
1123:
1105:
1100:
1097:
1092:
1089:
1086:
1074:
1073:vertical line
1068:
1059:
1053:
1047:
1045:
1040:
1030:
1026:
1022:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
972:
971:
970:
966:
962:
952:
948:
942:
935:
931:
925:
918:
916:
914:
913:
906:
901:
900:
893:
888:
884:
880:
876:
860:
855:
852:
847:
840:
819:
816:
811:
804:
800:
796:
792:
773:
768:
765:
760:
757:
752:
749:
744:
741:
738:
731:
730:
729:
700:
697:
694:
691:
688:
685:
682:
679:
676:
669:
668:
667:
663:
656:
648:
646:
635:
609:
606:
603:
600:
597:
594:
591:
588:
585:
582:
563:Two variables
562:
560:
544:
541:
536:
533:
530:
521:
507:
504:
501:
481:
478:
475:
472:
469:
466:
463:
447:
445:
443:
439:
435:
431:
426:
424:
420:
416:
411:
406:of dimension
405:
399:
394:
386:
382:
378:
374:
369:
367:
363:
347:
344:
339:
335:
325:
323:
318:
316:
312:
307:
291:
287:
283:
280:
277:
272:
268:
259:
255:
251:
247:
229:
225:
221:
218:
215:
210:
206:
202:
199:
191:
187:
169:
165:
161:
158:
155:
150:
146:
125:
122:
119:
116:
113:
108:
104:
98:
94:
90:
87:
84:
79:
75:
69:
65:
56:
52:
48:
39:
33:
19:
4003:Discriminant
3922:Multivariate
3842:
3762:
3743:
3723:
3703:
3680:
3673:
3660:
3653:
3626:
3614:
3564:function of
3557:real numbers
3554:
3413:, yielding
3408:
3404:
3396:
3392:
3389:
3381:affine space
3374:-dimensional
3360:
3341:
3327:
3323:inconsistent
3322:
3320:
3313:
3237:
3230:
3223:
3219:
3214:
3210:
3206:
3198:
3191:
3084:
3064:
3053:
2791:
2532:
2525:
2517:
2369:
2360:
2353:
2350:
2209:
2008:
2001:
1998:
1988:
1981:
1970:
1963:
1959:
1945:
1936:
1933:
1861:
1852:
1849:
1841:
1774:
1707:
1593:
1457:
1400:
1346:
1283:
1276:
1223:
1207:
1196:
1188:
1138:
1127:
1124:
1072:
1066:
1051:
1048:
1043:
1041:
1018:
964:
960:
956:
950:
946:
933:
929:
910:
904:
897:
891:
874:
788:
715:
661:
658:
636:
566:
522:
451:
448:One variable
430:real numbers
427:
412:
397:
384:
370:
365:
361:
326:
319:
308:
250:real numbers
246:coefficients
50:
44:
3952:Homogeneous
3947:Square-free
3942:Irreducible
3807:Polynomials
3648:, pp. 52-53
3561:real-valued
3215:coefficient
2528:determinant
1281:-intercept
1205:-intercept
912:linear maps
419:engineering
258:expressions
47:mathematics
4023:Categories
3912:Univariate
3696:References
3367:hyperplane
3056:hyperplane
1201:, and its
1136:, and, if
839:-intercept
393:hyperplane
313:over some
254:parameters
4034:Equations
3998:Resultant
3937:Trinomial
3917:Bivariate
3769:EMS Press
3498:≠
3480:…
3468:∈
3461:∑
3457:−
3437:−
3351:variables
3142:⋯
2893:−
2848:−
2813:−
2753:−
2728:−
2701:−
2683:−
2630:−
2608:−
2592:−
2576:−
2554:−
2474:−
2429:−
2394:−
2314:−
2292:−
2276:−
2260:−
2238:−
2179:−
2157:−
2132:−
2103:−
2061:−
2036:−
1814:−
1789:−
1744:−
1677:−
1557:−
1520:−
1483:−
1370:−
1316:−
1157:−
1093:−
848:−
812:−
761:−
745:−
630:not both
537:−
505:≠
402:) in the
345:≠
322:solutions
281:…
219:…
186:variables
159:…
88:…
3983:Division
3932:Binomial
3927:Monomial
3621:, pg. 15
3574:See also
3353:are the
879:calculus
791:function
244:are the
190:unknowns
184:are the
55:equation
3771:, 2001
3633:, p. 25
3337:-tuples
434:complex
415:physics
366:unknown
192:), and
3820:degree
3732:
3710:
3369:in an
3233:> 0
793:. The
385:linear
138:where
53:is an
3606:Notes
3385:plane
3325:(for
3228:with
1214:(the
803:slope
801:with
795:graph
626:and
622:with
494:with
438:field
315:field
3809:and
3730:ISBN
3708:ISBN
3379:(or
3363:− 1)
3342:The
3291:and
1978:and
1943:and
1859:and
1708:or
1035:and
1029:line
885:, a
834:and
799:line
641:and
571:and
417:and
381:line
320:The
188:(or
49:, a
3818:By
3400:≠ 0
3330:≠ 0
3067:– 1
2210:By
1999:If
1141:= 0
1130:≠ 0
1069:= 0
1060:of
1054:≠ 0
1049:If
907:= 0
894:= 0
664:≠ 0
659:If
637:If
520:.
400:− 1
45:In
4025::
3767:,
3761:,
3638:^
3570:.
3387:.
3235:.
3177:0.
3077:.
3039:0.
2777:0.
2359:=
2007:≠
1987:,
1969:,
1919:1.
1122:.
963:,
949:=
932:=
915:.
634:.
559:.
444:.
410:.
368:.
3799:e
3792:t
3785:v
3566:n
3540:.
3535:i
3531:x
3523:j
3519:a
3513:i
3509:a
3501:j
3495:i
3492:,
3489:}
3486:n
3483:,
3477:,
3474:1
3471:{
3465:i
3450:j
3446:a
3442:b
3434:=
3429:j
3425:x
3409:j
3405:x
3397:j
3393:a
3372:n
3361:n
3359:(
3349:n
3344:n
3335:n
3328:b
3316:n
3299:z
3279:y
3275:,
3272:x
3252:3
3249:=
3246:n
3231:i
3224:i
3220:a
3202:0
3199:a
3194:b
3174:=
3171:b
3168:+
3163:n
3159:x
3153:n
3149:a
3145:+
3139:+
3134:2
3130:x
3124:2
3120:a
3116:+
3111:1
3107:x
3101:1
3097:a
3065:n
3060:n
3036:=
3031:|
3025:1
3018:2
3014:y
3006:2
3002:x
2994:1
2987:1
2983:y
2975:1
2971:x
2963:1
2958:y
2953:x
2947:|
2922:0
2919:=
2916:)
2911:1
2907:y
2901:2
2897:x
2888:2
2884:y
2878:1
2874:x
2870:(
2867:+
2864:y
2861:)
2856:1
2852:x
2843:2
2839:x
2835:(
2832:+
2829:x
2826:)
2821:2
2817:y
2808:1
2804:y
2800:(
2774:=
2769:|
2761:1
2757:y
2748:2
2744:y
2736:1
2732:x
2723:2
2719:x
2709:1
2705:y
2698:y
2691:1
2687:x
2680:x
2674:|
2649:0
2646:=
2643:)
2638:1
2634:x
2627:x
2624:(
2621:)
2616:1
2612:y
2603:2
2599:y
2595:(
2589:)
2584:1
2580:y
2573:y
2570:(
2567:)
2562:1
2558:x
2549:2
2545:x
2541:(
2503:0
2500:=
2497:)
2492:1
2488:y
2482:2
2478:x
2469:2
2465:y
2459:1
2455:x
2451:(
2448:+
2445:y
2442:)
2437:1
2433:x
2424:2
2420:x
2416:(
2413:+
2410:x
2407:)
2402:2
2398:y
2389:1
2385:y
2381:(
2364:2
2361:x
2357:1
2354:x
2336:,
2333:0
2330:=
2327:)
2322:1
2318:x
2311:x
2308:(
2305:)
2300:1
2296:y
2287:2
2283:y
2279:(
2273:)
2268:1
2264:y
2257:y
2254:(
2251:)
2246:1
2242:x
2233:2
2229:x
2225:(
2195:.
2192:)
2187:1
2183:x
2176:x
2173:(
2165:1
2161:x
2152:2
2148:x
2140:1
2136:y
2127:2
2123:y
2116:=
2111:1
2107:y
2100:y
2077:.
2069:1
2065:x
2056:2
2052:x
2044:1
2040:y
2031:2
2027:y
2012:2
2009:x
2005:1
2002:x
1994:)
1992:2
1989:y
1985:2
1982:x
1980:(
1976:)
1974:1
1971:y
1967:1
1964:x
1962:(
1949:0
1946:y
1940:0
1937:x
1916:=
1909:0
1905:y
1901:y
1896:+
1889:0
1885:x
1881:x
1865:0
1862:y
1856:0
1853:x
1827:)
1822:1
1818:x
1811:x
1808:(
1805:m
1802:=
1797:1
1793:y
1786:y
1760:.
1755:1
1751:x
1747:m
1739:1
1735:y
1731:+
1728:x
1725:m
1722:=
1719:y
1693:,
1690:)
1685:1
1681:x
1674:x
1671:(
1668:m
1665:+
1660:1
1656:y
1652:=
1649:y
1624:1
1620:y
1616:,
1611:1
1607:x
1596:m
1570:.
1565:b
1562:c
1554:=
1545:0
1541:y
1533:,
1528:a
1525:c
1517:=
1508:0
1504:x
1496:,
1491:b
1488:a
1480:=
1473:m
1443:,
1440:0
1437:=
1434:c
1431:+
1428:y
1425:b
1422:+
1419:x
1416:a
1386:.
1381:0
1377:x
1373:m
1367:x
1364:m
1361:=
1358:y
1332:,
1329:)
1324:0
1320:x
1313:x
1310:(
1307:m
1304:=
1301:y
1287:0
1284:x
1279:x
1262:.
1257:0
1253:y
1249:+
1246:x
1243:m
1240:=
1237:y
1220:y
1216:y
1211:0
1208:y
1203:y
1199:m
1170:.
1165:b
1162:c
1154:=
1151:y
1139:a
1134:y
1128:a
1120:x
1106:,
1101:a
1098:c
1090:=
1087:x
1077:y
1067:b
1062:x
1052:b
1037:b
1033:a
1004:0
1001:=
998:c
995:+
992:y
989:b
986:+
983:x
980:a
967:)
965:y
961:x
959:(
951:b
947:y
934:a
930:x
905:c
892:c
861:.
856:b
853:c
837:y
820:b
817:a
774:.
769:b
766:c
758:x
753:b
750:a
742:=
739:y
726:y
722:x
718:y
701:0
698:=
695:c
692:+
689:y
686:b
683:+
680:x
677:a
662:b
643:b
639:a
632:0
628:b
624:a
610:,
607:0
604:=
601:c
598:+
595:y
592:b
589:+
586:x
583:a
573:y
569:x
545:a
542:b
534:=
531:x
508:0
502:a
482:,
479:0
476:=
473:b
470:+
467:x
464:a
454:x
408:n
398:n
389:n
348:0
340:1
336:a
292:n
288:a
284:,
278:,
273:1
269:a
230:n
226:a
222:,
216:,
211:1
207:a
203:,
200:b
170:n
166:x
162:,
156:,
151:1
147:x
126:,
123:0
120:=
117:b
114:+
109:n
105:x
99:n
95:a
91:+
85:+
80:1
76:x
70:1
66:a
34:.
20:)
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