KS3 · Maths

Equation of a line in general form

Every straight line has to meet the axes somewhere. There is one trick that finds both meeting points in seconds: put a zero in.

2y + 3x − 6 = 0: slide the point along the line

00.511.5201234xy(1, 1.5)

x: 1. y: 1.5

Slide me to the y-axis (the left edge), then to the x-axis (the bottom edge), and read my coordinates.

Two ways to write a straight line

y = mx + c or ay + bx + c = 0

In y = mx + c, m is the gradient (how much y goes up for every 1 across) and c is the y-intercept, because x = 0 gives y = c. In ay + bx + c = 0, everything sits on one side and equals zero: a is the number in front of y, b is the number in front of x, and c is the constant, if there is one.

Why use the second form? Put x = 0 or y = 0 into it and the intercepts drop out, which makes a quick sketch easy. For y = mx + c you plot from the y-intercept and gradient instead: for y = 4 − 3x, start at (0, 4), go right 1 and down 3, then draw the line all the way across with a ruler. Read the numbers on the axes rather than just counting squares.

Algebra · Straight lines

Same line, different costume

Each equation below is a costume worn by one of the four lines. Which line is it?

Still to sort

y = 2x − 3 (0)

y = 4x + 5 (0)

y = 3x − 1 (0)

y = 10 − 2x (0)

9 of 9 still to sort.

One straight line can be written in lots of ways. Work out which of four lines each equation really is.

When a and b are not 1

Problem

Find the intercepts of 2y + 3x − 6 = 0, the line from the graph, without drawing it. Then do the same for 2y + 5x − 20 = 0.

Algebra · Straight lines

Where does y + x + 8 = 0 cross the y-axis?

A friend looks at y + x + 8 = 0 and says, "You can just read it off the end."

Which is closest to what you think the y-intercept is, and why?
How sure are you?

Predict, then check

Picture the line first: every point on it has an x value of 3.

The vertical line x = 3 is a straight line. Can it be written as y = mx + c?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Where does a line meet the axes? Put a zero in and find out.

What you need to know

  • An equation of a line is any equation whose graph is a straight line, and the same line can be written in different forms.
  • In y = mx + c, m is the gradient and c is the y-intercept. In ay + bx + c = 0, everything is on one side and equals zero, with a coefficient of y, a coefficient of x and maybe a constant.
  • It does not matter which way round the x and y terms are written: 4x + 5 − y = 0, 4x − y + 5 = 0 and −y + 4x + 5 = 0 are all the same line as y = 4x + 5.
  • To find the y-intercept, substitute x = 0. To find the x-intercept, substitute y = 0. Then plot both points and join them with a straight line.
  • When a and b are not 1, you have to divide by the coefficient. The intercepts are just −c only when a and b are both 1.
  • Vertical lines, like x = 3, can't be written as y = mx + c.

The big picture

A straight line can be written in more than one way. Besides y = mx + c there is the form ay + bx + c = 0, where everything sits on one side and equals zero. In that form, putting x = 0 gives the y-intercept and putting y = 0 gives the x-intercept, so you can sketch the line from those two points. Read the answer by substituting and calculating, not by guessing from the constant, and remember that vertical lines like x = 3 are the one kind y = mx + c cannot describe.

Key points

1An intercept is a point on the line where one coordinate is zero: x = 0 on the y-axis, y = 0 on the x-axis.
2Substitute and calculate: for y + x + 8 = 0, putting x = 0 gives y + 8 = 0, so the y-intercept is (0, −8), not (0, 8).
3For 2y + 3x − 6 = 0 the intercepts are (0, 3) and (2, 0). The coefficient of y or x means a divide step.
4Two intercepts are enough to sketch a line: mark them and draw a straight line through them with a ruler.
5y = mx + c can't describe every straight line, because a vertical line has no y-intercept and no gradient you can write as a number.

Worked example

Problem

Sketch the line y + x − 4 = 0 by finding where it crosses the axes.

⚠ Watch out

Mixing up which zero goes where. For the y-intercept you substitute x = 0, not y = 0, because every point on the y-axis has x = 0. For the x-intercept you substitute y = 0. If you can't remember, picture where you are: on the y-axis, x is 0.

🧠

Memory hook

Put a zero in and you land on the OTHER axis: x = 0 puts you on the y-axis, and y = 0 puts you on the x-axis.

✓

Check yourself

Try it cold: find where 4y + 3x − 24 = 0 meets both axes, then put each point back into the equation. If the left side comes to 0, it is on the line.

Flashcards

(13)
What is an equation of a line?
Any equation whose graph forms a straight line.
In y = mx + c, what does m tell you?
The gradient: how much y goes up for every increase of one in x.
In y = mx + c, what is c, and why?
The y-intercept. When x is zero, y = c.
What does ay + bx + c = 0 look like in words?
Everything on one side, equal to zero, with a coefficient of y, a coefficient of x and maybe a constant.
Does the order of the x and y terms matter in ay + bx + c = 0?
No. 4x + 5 − y = 0, 4x − y + 5 = 0 and −y + 4x + 5 = 0 all describe the same line, y = 4x + 5.
How do you find the y-intercept from the equation?
Substitute x = 0 and solve for y.
How do you find the x-intercept from the equation?
Substitute y = 0 and solve for x.
Why does x = 0 give the y-intercept?
Every point on the y-axis has x = 0, so the line meets the y-axis where x is zero.
When are the intercepts just −c?
Only when a and b are both 1. Otherwise divide by the coefficient.
What is the y-intercept of y − x − 4 = 0?
Put x = 0: y − 4 = 0, so y = 4. The y-intercept is (0, 4).
How do you sketch a line from ay + bx + c = 0?
Find both intercepts, plot them, and draw a straight line through them with a ruler.
Which straight lines can't be written as y = mx + c?
Vertical lines, x = a. They never cross the y-axis, so there is no y-intercept, and the gradient can't be written as a number.
How can you check a point is on a line?
Put its x and y into the equation. For y = 2x + 4, the point (2, 8) works because 2 × 2 + 4 = 8.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

More KS3 Maths topics

See the full KS3 Maths curriculum →

How this lesson was checked. This KS3 Mathslesson was published through Lightbulb Learning's human-designed editorial process — the educational standards, accuracy rules and publication checks it must pass were authored and approved by Philip Halpin. It passed subject-specific assessment, automated educational checks and technical publication verification before going live (publication checks completed 2 October 2026). Published pages are monitored, human spot-checking is ongoing across the lesson library, and anything found wrong is corrected or withdrawn. How our lessons are made and checked. Spotted a mistake? Email hello@lightbulblearning.co and we'll review it.