Lesson Guide
Explanation
When an equation is written as y equals m x plus c, you can read two facts straight off it. The number multiplying x, called m, is the gradient. The number on its own, called c, is the y-intercept: where the line crosses the y-axis. For y equals two x minus three, the gradient is two and the line crosses the y-axis at negative three. On the graph, start at zero, negative three and go up two for every one across. Be careful when the terms are the other way round: y equals five minus x has gradient negative one and y-intercept five.
Your Turn
Read m and c for y = 7 − 3x. Check: m = −3, c = 7.
Watch Out
m is the number multiplying x, wherever it is written. In y = 5 − x, m is −1, not 5.
Explanation
Many lines are not written as y equals m x plus c. To read the gradient of three x plus two y equals eight, first make y the subject. Subtract three x from both sides: two y equals negative three x plus eight. Then divide every term by two: y equals negative three halves x plus four. Now you can read it: the gradient is negative three halves and the y-intercept is four. Check with the point zero, four: three times zero plus two times four is eight.
Your Turn
Rearrange x + 3y = 9. Check: y = −\(\dfrac{1}{3}\)x + 3.
Watch Out
Divide every term. y = −3x + 4 and y = −\(\dfrac{3}{2}\)x + 8 each divide only one term by 2.
Explanation
The y-intercept is on the y-axis, where x is zero. The x-intercept is on the x-axis, where y is zero. For two x minus five y equals ten, put x equals zero: negative five y equals ten, so y equals negative two. Put y equals zero: two x equals ten, so x equals five. The line crosses the axes at zero, negative two and five, zero. You do not need to rearrange the equation first: this works on any form.
Your Turn
Find the intercepts of 3x + 4y = 12. Check: x-intercept 4, y-intercept 3.
Watch Out
To find the y-intercept, put x = 0, not y = 0. Each intercept zeroes the OTHER letter.
Explanation
To sketch a line quickly, find where it crosses the two axes. For y equals negative two x plus six, put x equals zero to get the y-intercept, six. Put y equals zero: zero equals negative two x plus six, so x equals three. Plot zero, six and three, zero, and rule a line through them, extending past both. Check the gradient: from zero, six down to three, zero, the rise is negative six and the run is three, so the gradient is negative two, which matches the equation.
Your Turn
Find the intercepts of y = 3x − 6. Check: (0, −6) and (2, 0).
Watch Out
The x-intercept is on the x-axis, so its y-coordinate is 0. Do not plot the 3 on the y-axis.
Explanation
When the gradient is a fraction, use it as a staircase. For y equals two thirds x minus one, start at the y-intercept, zero, negative one. A gradient of two thirds means a run of three and a rise of two. From zero, negative one, go three across and two up to reach three, one. Do it again to reach six, three. Rule the line through the points. If the gradient is negative, go across to the right and then down. Quiz and exercises now.
Your Turn
Start y = −\(\dfrac{3}{4}\)x + 2 at (0, 2). Which lattice point do you reach next, to the right? Check: (4, −1).
Watch Out
For \(\dfrac{2}{3}\) the run is 3 and the rise is 2. Going across 2 and up 3 draws a gradient of \(\dfrac{3}{2}\).
📌 Key Points — this lesson in a nutshell
- In y = mx + c, m is the gradient and c is the y-intercept, where the line crosses the y-axis.
- Make y the subject before you read anything off: 3x + 2y = 8 becomes y = −\(\dfrac{3}{2}\)x + 4. Divide every term.
- y-intercept: put x = 0. x-intercept: put y = 0.
- To sketch, plot the two intercepts, or start at c and step out the gradient: run across, then rise up (or down if m is negative).
Step 1: Introduction
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