Linear and Non-Linear Simultaneous Equations

Question 1

Solve y = x + 1 and y = x² − 1. Which are the solution points?

Accepted Answer

(2, 3) and (−1, 0) — x² − 1 = x + 1, so x² − x − 2 = 0, (x − 2)(x + 1) = 0. x = 2, y = 3 or x = −1, y = 0.

Question 2

Solve x + y = 5 and xy = 6.

Accepted Answer

(2, 3) and (3, 2) — x and y are roots of t² − 5t + 6 = 0, i.e. 2 and 3, so (2, 3) and (3, 2).

Question 3

For what value of c is y = 2x + c a tangent to y = x²?

Accepted Answer

c = −1 — x² = 2x + c gives x² − 2x − c = 0. Tangent needs Δ = 4 + 4c = 0, so c = −1.

Question 4

How many points of intersection do y = x² + 1 and y = x have?

Accepted Answer

0 — x² + 1 = x gives x² − x + 1 = 0, Δ = 1 − 4 = −3 < 0, so no intersection.

Question 5

Solve y = 3 − x and x² + y² = 5.

Accepted Answer

(1, 2) and (2, 1) — x² + (3 − x)² = 5, so 2x² − 6x + 4 = 0, x² − 3x + 2 = 0, x = 1 or 2, giving (1, 2) and (2, 1).

Question 6

Solve y = x² − 4x + 5 and y = 2x − 3.

Accepted Answer

(2, 1) and (4, 5) — x² − 4x + 5 = 2x − 3, so x² − 6x + 8 = 0, (x − 2)(x − 4) = 0. x = 2, y = 1 or x = 4, y = 5.

Question 7

For which value of k does the line y = k meet y = x² − 3 exactly once?

Accepted Answer

k = −3 — x² − 3 = k gives x² = k + 3; one solution needs k + 3 = 0, so k = −3.

Question 8

Solve y = 2x and x² + y² = 20.

Accepted Answer

(2, 4) and (−2, −4) — x² + (2x)² = 20, so 5x² = 20, x² = 4, x = ±2, giving (2, 4) and (−2, −4).

Question 9

y = x² − 2x meets y = 3x − 4. Which point has the smaller x-coordinate?

Accepted Answer

(1, −1) — x² − 2x = 3x − 4, so x² − 5x + 4 = 0, (x − 1)(x − 4) = 0. At x = 1, y = −1.

Question 10

Find the values of m for which y = mx + 1 is a tangent to y = x² + x + 2.

Accepted Answer

m = −1 or 3 — x² + x + 2 = mx + 1 gives x² + (1 − m)x + 1 = 0. Δ = (1 − m)² − 4 = 0, so 1 − m = ±2, m = −1 or 3.