Simultaneous equations: linear and quadratic

The Maths Habit

  1. Solve y = 3x − 1 and y = x + 7

    x = 4, y = 11

  2. Solve 3x + y = 19 and x − y = 1

    x = 5, y = 4

  3. Solve 5x + 2y = 16 and 3x + 2y = 8

    x = 4, y = −2

A straight line and a U shaped curve are drawn on the same axes. How many times could they cross?

Twice: the line cuts through the curve

Once: the line just touches the curve

Never: the line misses the curve

xy−3−2−112342468100

Curve: y = x2
Line: y = x + 6

Where do the curve and the line cross?

(−2, 4) and (3, 9)

At a crossing point both graphs have the same y.

x−3−2−10123
y = x29410149
y = x + 63456789

Which x values give the same y? x = −2 and x = 3

Same y, so put them equal:

x2 = x + 6

x2 − x − 6 = 0

(x − 3)(x + 2) = 0

x = 3 or x = −2: the crossing points, with no graph to draw

Solve y = x2 and y = x + 2.

put them equal, then = 0x2 − x − 2 = 0factorise(x − 2)(x + 1) = 0solve
x =2or−1
Answer(2, 4) and (−1, 1)

Solve y = x2 + 1 and y = 2x + 4.

put them equal, then = 0x2 − 2x − 3 = 0factorise(x − 3)(x + 1) = 0solve
x =3or−1
Answer(3, 10) and (−1, 2)

Solve y = x2 and y = x + 2.

put them equal, then = 0x2 − x − 2 = 0factorise(x − 2)(x + 1) = 0solve
x =2or−1
Answer(2, 4) and (−1, 1)

Solve y = x2 + 3 and y = 4x.

put them equal, then = 0x2 − 4x + 3 = 0factorise(x − 1)(x − 3) = 0solve
x =1or3
Answer(1, 4) and (3, 12)

Simultaneous equations: linear and quadratic worksheet, page 1

Answer sheet

  1. Solve x2 + y2 = 25 and y = x − 1.

    (4, 3) and (−3, −4)

  2. The line y = 2x − 1 touches the curve y = x2 at one point. Find the point.

    (1, 1)

  1. Solve y = x2 and y = x + 12

    (4, 16) and (−3, 9)

  2. Solve y = x2 + 3x and y = 2x + 6

    (2, 10) and (−3, 0)

  3. The line y = 6x − 9 meets the curve y = x2. Find where. How many points are there?

    (3, 9), one point
Exit tickets to print