Chapter 7 of 8

All 8 chapters
  1. 1Coordinates, readunderstood
  2. 2All four quarters of the grid, readunderstood
  3. 3Shapes on a grid, readunderstood
  4. 4From a rule to a line, readunderstood
  5. 5Lines like x = 3 and y = −2, readunderstood
  6. 6Steepness: the gradient, readunderstood
  7. 7The equation y = mx + c, readunderstood
  8. 8Graphs from real life, readunderstood

M3.9.7 Chapter 7

The equation y = mx + c

Can I see how steep a line is, and where it crosses, from its rule alone?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find the gradient of y = 3x − 2 from its equation
  • Find where y = 3x − 2 crosses the y axis
  • Write the equation of a line from its graph

Every straight line that is not upright has a rule that looks like this: y = mx + c. The letters m and c stand for two numbers. They tell you almost everything about the line.

The number m is in front of x. It is the gradient, how steep the line is. The number c stands alone. It is where the line crosses the y axis. This crossing point is the y intercept, the point (0, c).

Draw it first

Draw the line y = 3x − 2 from m and c

  1. A grid with the x axis across and the y axis up. Step 1: the dot (0, −2) on the y axis, where c = −2. Step 2: a step triangle from that dot, marked run 1 and rise 3, to the point (1, 1). Step 3: the straight line y = 3x − 2 through the points.−2−11234−3−2−1123450xy

    1 The number on its own is c = −2. Plot the point (0, −2) on the y axis.

  2. A grid with the x axis across and the y axis up. Step 1: the dot (0, −2) on the y axis, where c = −2. Step 2: a step triangle from that dot, marked run 1 and rise 3, to the point (1, 1). Step 3: the straight line y = 3x − 2 through the points.run 1rise 3−2−11234−3−2−1123450xy

    2 The gradient is m = 3. That is 3 up for 1 across. From (0, −2) go 1 across and 3 up. Plot (1, 1).

  3. A grid with the x axis across and the y axis up. Step 1: the dot (0, −2) on the y axis, where c = −2. Step 2: a step triangle from that dot, marked run 1 and rise 3, to the point (1, 1). Step 3: the straight line y = 3x − 2 through the points.run 1rise 3−2−11234−3−2−1123450xy

    3 Draw the line through the two points. It is the line y = 3x − 2.

Worked example

Find the gradient of y = 3x − 2 and where it crosses the y axis.

  1. Compare the rule with y = mx + c. The number in front of x is 3, so m = 3.
  2. The number on its own is −2, so c = −2.
  3. The gradient is 3. The line crosses the y axis at (0, −2).

Answer: 3 and −2

Worked example

A line goes through (0, 1) and (2, 5). Write its equation.

  1. The line crosses the y axis at (0, 1), so c = 1.
  2. From (0, 1) to (2, 5) the run is 2 and the rise is 4.
  3. The gradient is 4 ÷ 2 = 2, so m = 2.
  4. Put m and c into y = mx + c. The equation is y = 2x + 1.

Answer: y = 2x + 1

A grid with the x axis across and the y axis up. A line through the dots A (0, 1) and B (2, 5). A step triangle from A shows the run 2 and the rise 4.ABrun 2rise 41234561234560xy
A is where the line crosses the y axis. The step triangle gives the gradient.

Worked example

A line goes through (0, 4) and (2, 0). Write its equation.

  1. The line crosses the y axis at (0, 4), so c = 4.
  2. From (0, 4) to (2, 0) the run is 2 and the rise is −4. The line goes down.
  3. The gradient is −4 ÷ 2 = −2, so m = −2.
  4. The equation is y = −2x + 4.

Answer: y = −2x + 4

A grid with the x axis across and the y axis up. A line going down to the right through the dots A (0, 4) and B (2, 0). A step triangle from A shows the run 2 and the rise −4.ABrun 2rise −41234561234560xy
The line goes down to the right, so m is negative.

Now see what c does. Two lines with the same m but different c have the same steepness. They never meet. We say they are parallel. Changing c moves a line up or down, and does not turn it.

A grid with the x axis across and the y axis up. Two parallel lines, labelled A and B. Line A is y = 2x + 1 and line B is y = 2x − 1. They are the same steepness and never meet.AB123456−3−2−11234560xy
Line A is y = 2x + 1. Line B is y = 2x − 1. Both have m = 2. They are parallel, and A is 2 above B.

See it three ways

With things

A tank holds 2 litres at the start. A tap adds 3 litres every minute.

As a picture

A grid with the x axis across and the y axis up. A graph with minutes across and litres up. The dots (0, 2), (1, 5) and (2, 8) lie on the line y = 3x + 2.y = 3x + 212345671234567890minlitres

In symbols

y = 3x + 2, where x is the minutes and y is the litres. The gradient 3 is the litres added each minute. The 2 is the litres at the start.

Try it

Draw a line from m and c

You need: squared paper, a ruler, a pencil.

  1. Draw the axes on squared paper, from −5 to 5.
  2. Take the rule y = 2x − 1. Plot (0, −1) on the y axis, because c = −1.
  3. The gradient is 2. From the dot go 1 across and 2 up. Plot the new dot. Do it again.
  4. Lay the ruler along the dots and draw the line. Check that (3, 5) is on it.

What you will see: You do not need a table of values. The number c gives the start. The number m gives each step. Putting x = 3 into the rule gives y = 5, so the line passes through (3, 5).

If you cannot try it, this is what happens.
A grid with the x axis across and the y axis up. The dots (0, −1), (1, 1), (2, 3) and (3, 5) on the line labelled y = 2x − 1.y = 2x − 1−2−112345−3−2−11234560xy

A wrong idea many people have

The line y = 3x − 2 crosses the y axis at 3.

Why it feels right: The 3 is the first number you see. So it seems to be the starting point.

The test that breaks it: The y axis is where x = 0. Put x = 0 into y = 3x − 2. Then y = 3 × 0 − 2 = −2. The line crosses at (0, −2), not at 3.

What is true: The number in front of x is the gradient. The number on its own is where the line crosses the y axis. In y = 3x − 2 they are 3 and −2.

Key words

y intercept
the point where a line crosses the y axis. In y = mx + c it is (0, c).
parallel
lines with the same gradient. They never meet.

Back to your guess

The answer is B: The line y = 3x − 2 crosses the y axis at the point (0, −2).

On the y axis x is 0. Put x = 0 into y = 3x − 2 and you get y = −2. So the line crosses at (0, −2). The number on its own, here −2, tells where the line crosses the y axis.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. What is the gradient of the line y = 3x − 2?

    Tap your answer.

    1. Not this one. That is where the line crosses the y axis. The gradient is the number in front of x. Go to M3.9.7. Try another.

    2. Not this one. You may have dropped the minus sign and picked the lone number. The gradient is in front of x. Go to M3.9.7. Try another.

    3. Yes, that is right. In y = mx + c the number in front of x is m, the gradient. Here it is 3.

    4. Not this one. The gradient is not always 1. Look at the number in front of x. Go to M3.9.6. Try another.

    See the answer

    3

    In y = mx + c the number in front of x is m, the gradient. Here it is 3.

    • If you chose A: That is where the line crosses the y axis. The gradient is the number in front of x. Go to M3.9.7.
    • If you chose B: You may have dropped the minus sign and picked the lone number. The gradient is in front of x. Go to M3.9.7.
    • If you chose D: The gradient is not always 1. Look at the number in front of x. Go to M3.9.6.

  2. Where does the line y = 3x − 2 cross the y axis?

    Tap your answer.

    1. Yes, that is right. On the y axis x = 0. The number on its own is −2, so the line crosses at (0, −2).

    2. Not this one. The 3 is the gradient. The lone number tells where it crosses. Go to M3.9.7. Try another.

    3. Not this one. You swapped the numbers. On the y axis the across number is 0. Go to M3.9.1. Try another.

    4. Not this one. You dropped the minus sign. The lone number is −2. Go to M3.9.2. Try another.

    See the answer

    (0, −2)

    On the y axis x = 0. The number on its own is −2, so the line crosses at (0, −2).

    • If you chose B: The 3 is the gradient. The lone number tells where it crosses. Go to M3.9.7.
    • If you chose C: You swapped the numbers. On the y axis the across number is 0. Go to M3.9.1.
    • If you chose D: You dropped the minus sign. The lone number is −2. Go to M3.9.2.

  3. Which equation is the line through A and B?

    A grid with the x axis across and the y axis up. A line through the dots A (0, −3) and B (2, 1).AB−5−4−3−2−112345−5−4−3−2−1123450xy
    Tap your answer.

    1. Not this one. The line crosses the y axis below the origin, at −3, not above it. Go to M3.9.7. Try another.

    2. Not this one. You swapped m and c. The number in front of x is the gradient. The lone number is where the line crosses the y axis. Go to M3.9.7. Try another.

    3. Not this one. You used the rise, 4, as the gradient. The gradient is the rise divided by the run: 4 ÷ 2. Go to M3.9.7. Try another.

    4. Yes, that is right. The line crosses the y axis at (0, −3), so c = −3. The rise is 4 and the run is 2, so m = 4 ÷ 2 = 2.

    See the answer

    y = 2x − 3

    The line crosses the y axis at (0, −3), so c = −3. The rise is 4 and the run is 2, so m = 4 ÷ 2 = 2.

    • If you chose A: The line crosses the y axis below the origin, at −3, not above it. Go to M3.9.7.
    • If you chose B: You swapped m and c. The number in front of x is the gradient. The lone number is where the line crosses the y axis. Go to M3.9.7.
    • If you chose C: You used the rise, 4, as the gradient. The gradient is the rise divided by the run: 4 ÷ 2. Go to M3.9.7.

  4. Which line is parallel to y = 4x + 1?

    Tap your answer.

    1. Not this one. You swapped m and c. Parallel lines have the same gradient, the number in front of x. This line has a different one. Go to M3.9.7. Try another.

    2. Yes, that is right. Parallel lines have the same gradient. The gradient of y = 4x + 1 is 4. Only y = 4x − 5 also has gradient 4.

    3. Not this one. Matching the lone number is not enough. Parallel lines have the same number in front of x. Go to M3.9.7. Try another.

    See the answer

    y = 4x − 5

    Parallel lines have the same gradient. The gradient of y = 4x + 1 is 4. Only y = 4x − 5 also has gradient 4.

    • If you chose A: You swapped m and c. Parallel lines have the same gradient, the number in front of x. This line has a different one. Go to M3.9.7.
    • If you chose C: Matching the lone number is not enough. Parallel lines have the same number in front of x. Go to M3.9.7.

  5. The line y = −x + 4 is drawn. What does it do?

    Tap your answer.

    1. Not this one. The number in front of x is −1. A negative gradient goes down to the right. Go to M3.9.6. Try another.

    2. Not this one. The −1 is the gradient. The lone number, 4, tells where it crosses the y axis. Go to M3.9.7. Try another.

    3. Yes, that is right. The gradient is −1, which is negative, so the line goes down. The lone number is 4, so it crosses at (0, 4).

    See the answer

    It goes down to the right and crosses the y axis at 4.

    The gradient is −1, which is negative, so the line goes down. The lone number is 4, so it crosses at (0, 4).

    • If you chose A: The number in front of x is −1. A negative gradient goes down to the right. Go to M3.9.6.
    • If you chose B: The −1 is the gradient. The lone number, 4, tells where it crosses the y axis. Go to M3.9.7.

  6. A line goes through (0, 4) and (2, 10). What is its gradient?

    See the answer

    3

    The run is 2 − 0 = 2. The rise is 10 − 4 = 6. The gradient is 6 ÷ 2 = 3.

  7. The line y = 4x − 6 crosses the y axis. What is its y number there?

    See the answer

    −6

    On the y axis x = 0. Put x = 0 into y = 4x − 6. Then y = −6.

  8. From an earlier chapter

    What is the gradient of the line through (2, 1) and (5, 10)?

    See the answer

    3

    The run is 5 − 2 = 3. The rise is 10 − 1 = 9. The gradient is 9 ÷ 3 = 3.

Keep it

Copy this into your exercise book.

y = mx + c. The number m in front of x is the gradient. The number c on its own is where the line crosses the y axis, at (0, c). Parallel lines have the same m.

Draw: Draw the line y = 2x + 1. Plot (0, 1). Go 1 across and 2 up. Plot a dot. Join them. Write m and c next to the line.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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