Teaching unit · UK Year 8-9 (Key Stage 3, ages 12 to 14)

Linear graphs and simultaneous equations

Reading gradient and intercept from y = mx + c, plotting and reading coordinates, and solving 2 linear equations together

About four lessons of 45 to 60 minutes

Student view

Start here · hook

Every straight line has a story: how steep, and where it starts

A phone plan costs a fixed £10 a month plus 5p per text. A taxi charges a £3 flag fall plus £2 per mile. A tank drains at a steady rate from a starting volume. Every one of these situations is a STRAIGHT LINE in disguise: a starting value (where the line crosses the y-axis) and a constant rate of change (how steeply it rises or falls). Writing y = mx + c captures both facts in one equation, m is the steepness (the gradient) and c is the starting value (the y-intercept).

Once 2 different situations are both written this way, comparing them (or finding exactly when they are equal) becomes a matter of algebra rather than guesswork. That is what solving SIMULTANEOUS equations does: it finds the 1 point where 2 different straight lines cross, the exact x and y that satisfy both equations at once.

Phone plan: cost = 10 + 0.05 x textsgradient 0.05 (5p per text), y-intercept 10 (the fixed monthly fee)
Taxi fare: cost = 3 + 2 x milesgradient 2 (£2 per mile), y-intercept 3 (the flag fall)
2 phone plans crossing on a graphthe crossing point is the exact number of texts where both plans cost the same, solvable by simultaneous equations
A table of x and y valuesplotting the points and joining them with a ruler reveals the same line the equation describes

Learning objective

What students will be able to do

Students will read the gradient and y-intercept from an equation in the form y = mx + c (rearranging first where needed), complete tables of values and find missing coordinates for a linear equation, and solve pairs of simultaneous linear equations using the elimination method, understanding the solution as the point where 2 lines cross.

Success criteria

I can state the gradient and y-intercept of a line directly from y = mx + c.
I can rearrange an equation like ax + y = k into the form y = mx + c.
I can complete a table of values for a linear equation, and find a missing x or y value.
I can find the gradient of a line from 2 known points on it.
I can solve a pair of simultaneous linear equations using the elimination method.

Curriculum anchor

Standards this unit teaches

KS3 Maths: AlgebraUK National Curriculum (England)
Algebra
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Algebra" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "work with coordinates in all 4 quadrants"; "recognise, sketch and produce graphs of linear...functions of 1 variable with appropriate scaling, using equations in x and y and the Cartesian plane"; "interpret mathematical relationships both algebraically and graphically"; "reduce a given linear equation in 2 variables to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically"; "use linear...graphs to estimate values of y for given values of x and vice versa and to find approximate solutions of simultaneous linear equations".

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Gradient: how steep a line is; the amount y increases (or decreases) for every 1 unit x increases, written as m in y = mx + c
y-intercept: the point where a line crosses the y-axis, where x = 0; written as c in y = mx + c
Linear equation: an equation whose graph is a straight line, with no powers of x higher than 1
Simultaneous equations: 2 (or more) equations that must be solved together, sharing the same solution for every variable
Elimination method: a way of solving simultaneous equations by adding or subtracting multiples of the equations to remove 1 variable
Coordinate plane: the grid formed by a horizontal x-axis and a vertical y-axis, used to plot points as (x, y)

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Reading gradient and y-intercept from y = mx + c

Concrete

Every equation in the form y = mx + c describes a straight line, and the 2 letters m and c tell you everything about it without drawing a single point. m is the GRADIENT, how much y changes for every 1 unit x increases (a bigger m means a steeper line, a negative m means the line goes downhill). c is the Y-INTERCEPT, the value of y when x = 0, exactly where the line crosses the y-axis.

If an equation is not already in the form y = mx + c, rearrange it first. For example, 3x + y = 8 rearranges (subtract 3x from both sides) to y = -3x + 8, so the gradient is -3 and the y-intercept is 8.

The line y = 2x + 3: it crosses the y-axis at (0, 3), so the y-intercept is 3, and y increases by 2 for every 1 that x increases, so the gradient is 2.

Worked example

State the gradient and y-intercept of y = -4x + 5, and of 2x + y = 9 (rearranged first).

y = -4x + 5 is already in the form y = mx + c: m = -4, c = 5.
2x + y = 9: subtract 2x from both sides to get y = -2x + 9. So m = -2, c = 9.

Answer: y = -4x + 5: gradient = -4, y-intercept = 5. 2x + y = 9: gradient = -2, y-intercept = 9.

Check for understanding, ask

What does a negative gradient tell you about the direction of a line?
State the gradient and y-intercept of y = 7x - 1.

2. Tables of values and finding missing coordinates

Pictorial

Every point (x, y) that makes an equation true lies on its line. A table of values is built by substituting several x values into the equation to find their matching y values, giving a set of coordinates that can be plotted and joined with a straight edge.

The same substitution works in reverse: if you are given a y value, rearrange to find x. For y = 3x - 2 with y = 10: 10 = 3x - 2, so 3x = 12, so x = 4.

Worked example

For y = 2x - 1, complete the table for x = -1, 0, 1, 2, then find x when y = 9.

x = -1: y = 2(-1) - 1 = -3. x = 0: y = 2(0) - 1 = -1. x = 1: y = 2(1) - 1 = 1. x = 2: y = 2(2) - 1 = 3.
For y = 9: 9 = 2x - 1, so 2x = 10, so x = 5.

Answer: Table: y = -3, -1, 1, 3. When y = 9, x = 5.

Check for understanding, ask

Why do all the points from a table for the same equation lie in a straight line?
For y = x + 6, find y when x = -2.

3. Finding the gradient from 2 points

Pictorial

If an equation is not known but 2 points on the line are, the gradient can still be found: it is the change in y divided by the change in x between the 2 points, often written as 'rise over run'.

Gradient between (0, 1) and (4, 5) = rise / run = (5 - 1) / (4 - 0) = 4 / 4 = 1.

Worked example

Find the gradient of the line through (1, 2) and (5, 14).

gradient = (y2 - y1) / (x2 - x1) = (14 - 2) / (5 - 1) = 12 / 4 = 3.

Answer: gradient = 3.

Check for understanding, ask

Why does it not matter which point you call (x1, y1) and which you call (x2, y2)?

4. Solving simultaneous equations by elimination

Abstract

2 different linear equations, each with its own line, cross at exactly 1 point (unless the lines are parallel). That crossing point's (x, y) is the ONLY pair of values that satisfies BOTH equations at once, which is exactly what 'solving simultaneous equations' means. The elimination method finds it algebraically: multiply the equations so 1 variable has matching coefficients, then add or subtract to eliminate it.

After eliminating 1 variable and solving for the other, substitute that value back into either ORIGINAL equation to find the 2nd variable. Always check the final answer in BOTH original equations.

Line 1: x + y = 8 (rearranged to y = -x + 8).

Line 2: y = x + 2. The 2 lines cross where both are true at once: at x = 3, y = 5 (check: 3 + 5 = 8, and 5 = 3 + 2), the algebraic solution to the simultaneous equations x + y = 8 and y = x + 2.

Worked example

Solve the simultaneous equations 2x + y = 11 and x + y = 7.

The y-coefficients already match (both 1), so subtract the 2nd equation from the 1st to eliminate y: (2x + y) - (x + y) = 11 - 7, giving x = 4.
Substitute x = 4 into the 2nd equation: 4 + y = 7, so y = 3.
Check in the 1st equation: 2(4) + 3 = 8 + 3 = 11. Correct.

Answer: x = 4, y = 3.

Check for understanding, ask

Why must the final (x, y) pair be checked in BOTH original equations?
Solve 3x + y = 10 and x + y = 4.

Watch for

Common misconceptions and how to address them

MisconceptionThe y-intercept is always a positive number, or is wherever the line 'starts' on the left of the graph.

Why it happens: Students confuse 'where the line starts' (a visual, left-edge impression) with the precise algebraic definition (the value of y exactly when x = 0).

How to address it: Always find the y-intercept by substituting x = 0, not by looking at the edge of a printed grid. Practise with equations that have a negative c, like y = 2x - 5.

MisconceptionA steeper line always has a bigger gradient number.

Why it happens: Steepness is judged visually, but a very negative gradient (like -10) also produces a steep line, just sloping the other way.

How to address it: Separate 2 questions: 'how steep' (the SIZE of m, ignoring its sign) and 'which way' (the SIGN of m: positive means uphill left to right, negative means downhill).

MisconceptionTo solve simultaneous equations, just solve each equation separately for x, then separately for y.

Why it happens: Each individual equation on its own has infinitely many (x, y) solutions (every point on that line); only when BOTH equations are used together does a single, unique solution emerge.

How to address it: Show a table of 3 different (x, y) pairs that satisfy equation 1 alone (all different), then check that only 1 of them ALSO satisfies equation 2. That 1 pair is the real, unique simultaneous solution.

Do it together

Guided practice (with answers)

1. State the gradient and y-intercept of y = 6x - 2.
Answer: gradient = 6, y-intercept = -2.
2. Rearrange 5x + y = 12 into y = mx + c, and state the gradient and y-intercept.
Answer: y = -5x + 12; gradient = -5, y-intercept = 12.
3. For y = 3x + 1, find y when x = 4.
Answer: 13, because y = 3(4) + 1 = 13.
4. Find the gradient of the line through (2, 3) and (6, 11).
Answer: 2, because (11 - 3) / (6 - 2) = 8 / 4 = 2.
5. Solve the simultaneous equations x + y = 10 and x - y = 2.
Answer: x = 6, y = 4, because adding the equations gives 2x = 12, so x = 6, then 6 + y = 10 gives y = 4.
6. Solve the simultaneous equations 2x + y = 9 and 3x + y = 13.
Answer: x = 4, y = 1, because subtracting the 1st from the 2nd gives x = 4, then 2(4) + y = 9 gives y = 1.

On their own

Independent practice worksheets

Practise gradient/intercept, tables of values, and solving simultaneous equations with computed, never-wrong answer keys.

Year 8 linear graphs: gradient & intercept worksheetthe core skill from section 1 Year 8 linear graphs: tables of values worksheetthe table and missing-coordinate skill from section 2 Year 9 simultaneous equations worksheetthe elimination-method skill from section 4

Reach every student

Differentiation

Support

For gradient/intercept, always underline the number multiplying x (gradient) and circle the number on its own (y-intercept) before answering.
For tables of values, do 1 column at a time and say the substitution out loud: 'x is negative 1, so y is 2 times negative 1, minus 1'.
For simultaneous equations, start with pairs where 1 variable's coefficients already match exactly (no multiplying needed) before introducing the multiply-first cases.
Keep a small reference card showing the 4 steps of elimination in order.

Extension

Solve a simultaneous-equations word problem (e.g. 2 ticket prices from 2 different combined-purchase totals).
Investigate what happens when 2 linear equations have the SAME gradient (no crossing point, no solution).
Find the equation of a line given only 2 points on it (combine the gradient-from-points and y-intercept skills).
Solve a simultaneous-equations pair where BOTH equations need multiplying by a different number before elimination.

Check it stuck

Assessment: exit ticket

A three-question exit ticket sampling gradient/intercept, tables, and simultaneous equations.

1. State the gradient and y-intercept of 4x + y = 20 (rearranged first).
Answer: y = -4x + 20; gradient = -4, y-intercept = 20.
2. For y = 5x - 3, find x when y = 17.
Answer: x = 4, because 17 = 5x - 3 gives 5x = 20, so x = 4.
3. Solve the simultaneous equations 3x + y = 14 and x + y = 6.
Answer: x = 4, y = 2, because subtracting gives 2x = 8, so x = 4, then 4 + y = 6 gives y = 2.

For the teacher

Teacher notes and timings

Rough timing across 4 lessons: Lesson 1 gradient/intercept (section 1), Lesson 2 tables and coordinates (section 2), Lesson 3 gradient from 2 points (section 3), Lesson 4 simultaneous equations (section 4) plus the exit ticket.
Every simultaneous-equations pair in the matching worksheet is constructed BACKWARD from a chosen integer solution (x0, y0), so the elimination method is guaranteed to reproduce it exactly by algebra, never by a rounded approximation.
The 2 functionGraph figures in section 4 each show 1 line only (the site's functionGraph figure kind draws a single connected line); the crossing point is stated algebraically in the caption rather than overlaid, since no figure kind currently draws 2 lines on 1 set of axes.
This unit deliberately covers only linear graphs (not quadratic graphs, piecewise, exponential or reciprocal graphs, which the real KS3 curriculum also lists but which sit closer to GCSE and are not a good fit for a computed, never-wrong printable worksheet at this stage).

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