Non-linear simultaneous equations
In a nutshell
Simultaneous equations can be solved where one of the equations is non-linear. This means that the one or both variables have powers. Questions will involve solving a pair of simultaneous equations where one equation is quadratic and the other is linear, or where one is the equation of a circle and the other is linear. Non-linear simultaneous equations can be solved using the substitution method.
Solve simultaneous equations with quadratic and linear equations
A quadratic equation of the form y=ax2+bx+c can be solved simultaneously with a linear equation ax+by=c by substituting the linear equation into the quadratic equation.
PROCEDURE
1.
| Rearrange the linear equation for either x or y. Label this as equation 1, label the quadratic equation 2. |
2.
| Substitute equation 1 into equation 2 to obtain a quadratic. |
3.
| Solve the quadratic, there should usually be two answers. |
4.
| Substitute the answers from step 3 into equation 1 to find two answers for the other variable. |
5.
| Check both pairs of answers in equation 2. |
Example 1
Solve simultaneously
y−7x=10y=x2+4x+6
Rearrange the linear equation and label both equations.
yy=7x+10=x2+4x+61◯2◯
Substitute equation 1 into equation 2 and solve.
7x+1000=x2+4x+6=x2−3x−4=(x−4)(x+1)x=4,x=−1
Substitute these answers for x into equation 1 to find the answers for y.
x=−1y=7x+10y=7×−1+10y=3x=4y=7x+10y=7×4+10y=38
Check both pairs of answers in equation 2.
yx=−1,y=3yx=4,y=38y=x2+4x+6=(−1)2+4(−1)+6=3=42+4(4)+6=38
The answers are
x=−1,y=3 or x=4,y=38
Solve simultaneous equations with circle and linear equations
A circle equation of the form x2+y2=r2 can be solved simultaneously with a linear equation ax+by=c by substituting the linear equation into the circle equation.
PROCEDURE
1.
| Rearrange the linear equation for either x or y. Label this as equation 1, label the circle equation 2. |
2.
| Substitute equation 1 into equation 2 to obtain a quadratic. |
3.
| Solve the quadratic, there should usually be two answers. |
4.
| Substitute the answers from step 3 into equation 1 to find two answers for the other variable. |
5.
| Check both pairs of answers in equation 2. |
Example 2
Solve simultaneously
y−x=−2x2+y2=20
Rearrange the linear equation and label both equations.
yx2+y2=x−2=201◯2◯
Substitute equation 1 into equation 2 and solve.
x2+(x−2)2x2+(x−2)(x−2)x2+x2−2x−2x+42x2−4x+42x2−4x−16x2−2x−8(x+2)(x−4)=20=20=20=20=0=0=0x=−2,x=4
Substitute these answers for x into equation 1 to find the answers for y.
x=−2yyyx=4yyy=x−2=−2−2=−4=x−2=4−2=2
Check both pairs of answers in equation 2.
x2+y2x=−2,y=−4(−2)2+(−4)2x=4,y=242+22=20=20=20
The answers are
x=−2,y=−4 or x=4,y=2