2.8 Solution of simultaneous equations 55
or if the same quadratic factor occurs mtimes,
(2.35)
For example (see Example 2.22c),
2.8 Solution of simultaneous equations
Consider the pair of linear equations
(1)x 1 + 1 y 1 = 13
(2)x 1 − 1 y 1 = 11
Equation (1) defines yas one function of x
y 1 = 131 − 1 x
whereas equation (2) defines yas a second function of x
y 1 = 1 x 1 − 11
Figure 2.14 shows that the graphs of these linear functions cross at the point
(x,y) 1 = 1 (2, 1); the two equations have the simultaneous solutionx 1 = 12 ,y 1 = 11.
In general, an algebraic equation in two variables xand ydefines either variable as an
algebraic function of the other. For example, the equation
p(x)y
n1 + 1 q(x)y
n− 11 +1-1+ 1 u(x)y 1 + 1 v(x) 1 = 10 (2.36)
defines one function y 1 = 1 f(x). A second algebraic equation,
p′(x)y
m1 + 1 q′(x)y
m− 11 +1-1+ 1 u′(x)y 1 + 1 v′(x) 1 = 10 (2.37)
1
342
1
122
1
1
1
2
32 2 2xxx x xx
x
x
−+− xx
=
−−+
=
−
−
()( ) −+ 22
ax b
xpxq
ax b
xpxq
ax b
x
11 mm222222++
++
++
()(+
ppx q
m- )
..............................................................................y=x− 1
y=3 −x
- 1
2
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Figure 2.14