7.2 One linear equation and one equation of order two in two unknowns
An equation of order two is an equation that in at least one term the powers of the variables add to exactly 2 and in no term the powers of the variables add to more than 2.
Examples: x 2 + y 2 = 4, x 2 − y = 7x, xy + x 2 = 2 y 2
The technique for solving one linear equation and one equation of order two is substitution.
ACTIVITY ACTION CTION
Using simultaneous equations involving a linear and an equation of order two
OBJECTIVE To learn the techniques needed to solve linear and order two simultaneous equations
Y 9 WORKED EXAMPLE
Solving linear and order two simultaneous equations
S Solve x − y = −3 and x 2 + xy = −1 simultaneously.
Step 1: Start with the linear equation x − y = −3 and solve for x or y. Choose cleverly. y = (x + 3) … (1)
Put a bracket around the terms on the right-hand side.
Step 2: Substitute this expression for y into x 2 + x( y) = −1 … (2) x 2 + x(x + 3) = −1
Multiply out the brackets: x 2 + x 2 + 3x = −1 2 x 2 + 3x + 1 = 0 [This is a quadratic in x.] Step 3: Solve the quadratic by factorisation or by using the quadratic formula: (2x + 1)(x + 1) = 0 x = − 1
_ 2 , x = −1
Step 4: Get the corresponding values for y using equation (1). x = − 1 x = −1:
_ 2 : y =
( − 1 _
2 , 5 _
2
( − 1 _
2 + 3
) = 5 _
2 y = (−1 + 3) = 2
) and (−1, 2) are the solutions. What do these solutions mean?
They are the points of intersection of the straight line with equation x − y = −3 and the curve with equation x 2 + xy = −1.
