GCSE Maths Linear equations & graphs questions by topic

Solve each equation.

1. (a)Solve 3x + 5 = 20.[1]
2. (b)Solve 2x − 7 = 9.[1]
3. (c)Solve 4x + 1 = x + 13.[2]

Tables and lines.

1. (a)Complete the table of values for y = 2x − 5 when x = 0, 1, 2, 3, 4.[2]
2. (b)Write down the gradient and the y-intercept of y = 2x − 5.[2]

A taxi charges a flag fall plus a per-mile rate.

1. (a)A taxi ride costs F = 3 + 1.8m pounds for m miles. State the flag fall and the cost per mile.[2]
2. (b)Work out the cost of a 7-mile ride.[2]

Reading solutions from a graph.

1. (a)Plot the lines y = x + 1 and y = 4 − x on the same axes for x from 0 to 4.[2]
2. (b)Write down the approximate solution of x + 1 = 4 − x from your graph, then check it algebraically.[2]

A phone contract charges by the gigabyte.

1. (a)The monthly bill in pounds is C = 10 + 2.5g, where g is the data used in GB. Work out the bill for 8 GB.[2]
2. (b)The bill for one month was £42.50. Work out the data used.[2]

Four equations are solved.

1. (a)Solve 3x + 8 = 2x + 17.[1]
2. (b)Solve 5(y − 2) = 30.[2]
3. (c)Solve (w + 4)⁄3 = 6.[1]

Equations with brackets and fractions.

1. (a)Solve 2(3x − 1) = 5x + 9.[3]
2. (b)Solve (4x − 3)⁄5 = 3.[2]

Equations from two points.

1. (a)Find the gradient of the line through (2, 5) and (6, 11).[2]
2. (b)Find the full equation of the line.[2]
3. (c)Does the point (10, 17) lie on the line? Justify your answer.[1]

Parallel lines in context.

1. (a)Line A has equation y = 4x − 3. Write down the equation of a parallel line passing through (0, 5).[2]
2. (b)Explain why the two lines never meet.[2]

A letter on both sides with fractions.

1. (a)Solve (2x + 1)⁄3 − x = 2.[3]
2. (b)Check your solution by substitution.[1]

A line is drawn through two points.

1. (a)The line passes through (2, 5) and (8, 17). Work out its gradient.[2]
2. (b)Write down the equation connecting x and y.[2]
3. (c)Test whether the point (30, 61) sits on your line.[1]

Perpendicular lines. (Higher)

1. (a)Find the gradient of any line perpendicular to y = 3x + 1, giving a reason.[2]
2. (b)Find the equation of the perpendicular line through (6, 1).[3]
3. (c)Verify that your line meets y = 3x + 1 at right angles by multiplying the gradients.[1]

A geometric application. (Higher)

1. (a)A(1, 2) and B(5, 10) lie on line L. Find the equation of L.[3]
2. (b)Find the equation of the line perpendicular to L that passes through B.[3]

Forming and solving from a context. (Higher)

1. (a)Two gyms charge: Gym A costs 30 + 4v pounds for v visits; Gym B costs 12 + 6v pounds. After how many visits do they cost the same?[3]
2. (b)Which gym is cheaper for 15 visits?[2]
3. (c)Interpret the gradients of the two cost equations.[1]

A mixed investigation. (Higher)

1. (a)P(2, 3) and Q(8, −3) lie on line M. Find the equation of M.[3]
2. (b)Find the equation of the line perpendicular to M through the midpoint of PQ.[3]
3. (c)Write down the coordinates of the point where your perpendicular line crosses the x-axis.[1]

A perpendicular line is constructed through a point. (Higher)

1. (a)The line l has equation y = 2x − 4. Write down the gradient of any line perpendicular to l.[1]
2. (b)Find the equation of the line perpendicular to l that passes through (6, 2).[3]
3. (c)Find the x-coordinate where your line crosses the x-axis.[1]

Three points form a triangle. (Higher)

1. (a)The points are A(1, 2), B(5, 10) and C(7, 9). Show that angle ABC is a right angle.[3]
2. (b)Show that AB = 4√5 and BC = √5.[2]
3. (c)Hence work out the area of triangle ABC.[1]

Two gym plans are compared.

1. (a)Plan P costs £15 per month plus £4 per visit. Work out the monthly cost of plan P for 6 visits.[1]
2. (b)Plan Q costs £35 per month with unlimited visits. Find the number of visits for which plans P and Q cost the same.[2]
3. (c)Dan expects 8 visits in a month. Work out which plan is cheaper for Dan, and by how much.[2]