One Linear, One Non-linear Revision Notes for Leaving Cert Mathematics

Simultaneous Equations - One Linear, One Non-linear

In this case of simultaneous equations, we will be faced with solving a pair of equations in which one equation is linear and one is non-linear.

Consider these cases :

We notice two points of intersection for each of the diagrams.

Example

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Solve the pair of simultaneous equations :

x + 2y = 8 \quad \text{(Equation A)} \\ x^2 + y^2 = 52 \quad \text{(Equation B)}

Solve for one variable :

x + 2y = 8

x = 8 - 2y

Substitute into the other equation :

x^2 + y^2 = 52

(8 - 2y)^2 + y^2 = 52

Expand and simplify :

64 - 32y + 4y^2 + y^2 = 52

64 - 32y + 5y^2 = 52

5y^2 - 32y + 12 = 0

Solve :

y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

y = \frac{-(-32) \pm \sqrt{(-32)^2 - 4(5)(12)}}{2(5)}

y = \frac{32 + 28}{10} = \frac{60}{10} = :success[6]

y = \frac{32 - 28}{10} = \frac{4}{10} = :success[0.4]

Solve for $x$ :

x = 8 - 2y

x = 8 - 2(6) = 8 - 12 = :success[-4]

x = 8 - 2(0.4) = 8 - 0.8 = :success[7.2]

Solutions :

(x, y) = :success[(-4, 6)] \quad \text{and} \quad :success[(7.2, 0.4)].