Year 9 · Unit 3 Linear relationships · Lesson 5 of 7

ax + by = d: the form that shows the intercepts

General form hides the gradient but shows the intercepts quickly. Change a, b and d, read both intercepts, and convert to y = mx + c.

FunctionsNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Lesson 3) State the gradient and y-intercept of y = −2x + 5.
    AnswerGradient −2; y-intercept 5.
  2. Solve 3x = 12.
    Answerx = 4.
  3. Rearrange 2y = 6 − 4x to give y.
    Answery = 3 − 2x: divide every term by 2.
Words used in this lesson
general form
a line written as ax + by = d. For example: 3x + 2y = 12.
intercept
where a line crosses an axis. For example: (4, 0) on the x-axis.
coefficient
the number multiplying a letter. For example: the 3 in 3x.
rearrange
change an equation so a different letter is on its own. For example: 2y = 6 − x becomes y = 3 − x/2.
constraint
a limit on what can be chosen. For example: a budget of 600 pesos.

Spending a fixed budget

At a market a stall sells mangoes for 40 pesos each and papayas for 60 pesos each. A student has exactly 600 pesos and will spend all of it. If x is the number of mangoes and y the number of papayas, then 40x + 60y = 600.

Predict first: what is the most mangoes she could buy, and the most papayas? Those two extremes are points on the graph. In the figure, set a, b and d and see how the extremes appear in any equation of the form ax + by = d.

2
3
6
2x + 3y = 6general form
3x-intercept (y = 0)
2y-intercept (x = 0)
−2/3gradient

2x + 3y = 6: with y = 0, x = 6 ÷ 2 = 3; with x = 0, y = 6 ÷ 3 = 2. The gradient is −2/3, so y = −2x/3 + 2.

Show me the steps
  1. 2x + 3y = 6: y = 0 gives x = 3; x = 0 gives y = 2. Gradient −2/3: y = −2x/3 + 2.
  2. x − y = 4: y = 0 gives x = 4; x = 0 gives y = −4. Gradient 1: y = x − 4.
  3. 4x = 8 (b = 0): x = 2 for every y. A vertical line with no y-intercept and no gradient.
  4. x + 2y = 0: both intercepts are 0, so the line passes through the origin. Gradient −1/2.
  5. 3x − 2y = −6: y = 0 gives x = −2; x = 0 gives y = 3. Gradient 3/2: y = 3x/2 + 3.

General form writes a line as ax + by = d. To find where it crosses the x-axis, put y = 0 and solve: x = d ÷ a. To find where it crosses the y-axis, put x = 0: y = d ÷ b.

To get y = mx + c, make y the subject: by = −ax + d, so y = (−a ÷ b)x + d ÷ b. The gradient is −a ÷ b, and the y-intercept c is d ÷ b. The constant d is not the c of y = mx + c.

When b = 0 there is no y term, and the line is vertical: x = d ÷ a. It has no gradient and cannot be written as y = mx + c.

Some countries call this form standard form. Here, as in the IB Diploma, standard form means scientific notation (Unit 1), so this form is called general form.

A common mistake: mixing up which variable to set to zero. The x-intercept is a point on the x-axis, where y is 0. For 2x + 5y = 20, put y = 0 to get x = 10, not x = 4.

Worked example: the market budget

Draw 40x + 60y = 600, with x (mangoes) across and y (papayas) up, and find the gradient.

  1. Put y = 0: 40x = 600, so x = 15. No papayas means 15 mangoes: the point (15, 0).
  2. Put x = 0: 60y = 600, so y = 10. No mangoes means 10 papayas: the point (0, 10).
  3. Join the points. The gradient is the rise over the run: (0 − 10) ÷ (15 − 0) = −2/3, which matches −a ÷ b = −40 ÷ 60. Every 3 mangoes given up buys 2 papayas.

Your go. The first step is given. Find the intercepts and gradient of 4x − 3y = 12.

  1. Put y = 0: 4x = 12, so x = 3. The x-intercept is (3, 0).
  2. Now put x = 0 to find the y-intercept, then find the gradient and check it against −a ÷ b.
Check the answer

x = 0 gives −3y = 12, so y = −4: the point (0, −4). The gradient is (−4 − 0) ÷ (0 − 3) = 4/3, and −a ÷ b = −4 ÷ (−3) = 4/3.

Check before you go on

1. For the line 3x + 4y = 12, what are the x-intercept and the y-intercept?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. What is the y-intercept of the line 3x + 5y = 30? Give the value of y.

3. Write x + 2y = 6 as y = mx + c. What is c?

Level 2 · Building

4. What is the gradient of 3x − 2y = 12? Give it as a decimal.

5. Where does 4x + 3y = −24 cross the x-axis? Give the value of x.

6. A buyer spends exactly 6000 shillings on beans at 300 shillings a bag and rice at 200 shillings a bag, so 300b + 200r = 6000. If he buys 10 bags of beans, how many bags of rice does he buy?

Level 3 · Stretch

7. The line 5x − 2y = 20 and the two axes enclose a triangle with the origin. What is its area, in square units?

8. A line has gradient 2/3 and crosses the y-axis at −4. Which is its equation in general form?

Something to think about

The lines 2x + 3y = 6 and 4x + 6y = 24 are drawn together. What do you notice about their intercepts and gradients? What is true of any two lines ax + by = d and ax + by = e?

One way to think about it

The first has intercepts 3 and 2; the second 6 and 4. Both have gradient −2/3, so they are parallel. Any two lines with the same a and b but different right-hand sides have the gradient −a/b in common, so they are parallel.

Exit ticket

9. A line crosses the y-axis at (0, 6) and the x-axis at (4, 0). Which equation does it satisfy?