GCSE Maths Simultaneous & inequalities questions by topic

Linear inequalities.

1. (a)Solve 4x − 3 < 13.[1]
2. (b)Solve −3x ≥ 12.[2]
3. (c)Show x > 1 on a number line.[1]

Elimination without scaling.

1. (a)Solve x + y = 7 and x − y = 3.[2]
2. (b)Check your solution in both equations.[2]

Substitution.

1. (a)Solve y = 2x and 3x + y = 15.[3]
2. (b)Write down the coordinates of the point where the lines y = 2x and 3x + y = 15 cross.[1]

A compound inequality.

1. (a)Solve 4 ≤ 3x − 2 < 10.[3]
2. (b)List the integers in the solution set.[1]

A cinema sells adult and child tickets.

1. (a)Adult tickets cost £8 and child tickets cost £5. One screening sold 90 tickets for £600. Write down two equations connecting a (adults) and c (children).[2]
2. (b)Solve your equations to find how many adult tickets were sold.[2]
3. (c)Check your answer in the other equation.[1]

Two inequalities are solved.

1. (a)Solve 4x − 3 < 13.[2]
2. (b)Solve −4 ≤ 2x < 6 and list the integer values of x.[2]

Elimination with scaling.

1. (a)Solve 2x + 3y = 12 and 3x + 4y = 17.[4]
2. (b)Interpret the solution as the crossing point of the two lines.[1]

Forming simultaneous equations.

1. (a)Two cinema tickets and one popcorn cost £19; one ticket and one popcorn cost £11. Write two equations and find the price of a ticket.[4]

Approximate solutions from a graph. (Higher)

1. (a)Plot y = 2x + 1 and y = 6 − x for x from 0 to 4.[2]
2. (b)Write down the approximate solution of 2x + 1 = 6 − x and check it algebraically.[2]

Inequalities with negatives.

1. (a)Solve 10 − 3x > 1.[2]
2. (b)Solve −4 ≤ 2x < 6 and list the integers.[2]

Scaling makes the coefficients match.

1. (a)Solve 4x + 3y = 20 and 3x + 4y = 22.[4]
2. (b)Check your solution in the equation you did not use last.[1]

Linear–quadratic systems. (Higher)

1. (a)Solve y = x + 2 and x² + y = 11.[3]
2. (b)Interpret your solutions as crossing points of the line and the curve.[1]
3. (c)Verify one pair by substitution into the quadratic.[2]

Quadratic inequalities. (Higher)

1. (a)Solve x² − 3x − 10 < 0.[3]
2. (b)Solve x² ≥ 9.[2]
3. (c)State whether x = 0 satisfies x² − 3x − 10 < 0, with a reason.[1]

Regions and set notation. (Higher)

1. (a)Shade and label the region R satisfying y ≥ x − 1, y ≤ 5 and x ≥ −2.[3]
2. (b)Write down the coordinates of the corners of R.[2]
3. (c)State whether the boundary lines are dashed or solid, with reasons.[1]

A full system. (Higher)

1. (a)Solve 4x + 3y = 5 and 2x − 5y = 9.[4]
2. (b)Hence solve the inequality pair 4a + 3b ≥ 5 and 2a − 5b ≤ 9 at the boundary point found, stating whether (x + 0.1, y) satisfies both.[3]

A line crosses a parabola twice. (Higher)

1. (a)Show that the intersection of y = x² − 3x + 7 and y = 2x + 3 satisfies x² − 5x + 4 = 0.[2]
2. (b)Write down the coordinates where the line and the parabola meet.[3]

A thrown ball stays above a height. (Higher)

1. (a)The height in metres is h = 15t − 5t² after t seconds. Show that h > 10 can be written as t² − 3t + 2 < 0.[2]
2. (b)Solve t² − 3t + 2 < 0.[3]
3. (c)Interpret your answer in the context of the flight.[1]

A region is bounded by four lines. (Higher)

1. (a)The region R satisfies x ≥ 0, x ≤ 3, y ≥ 1 and y ≤ x + 2. Write down the coordinates of all four corner points of R.[3]
2. (b)Determine whether the point (2, 4) lies in R.[1]
3. (c)A point (a, b) in R has a = 2. List the possible integer values of b.[2]