Year 10 Standard · Unit 2 Generalizing and coordinate geometry · Lesson 5 of 6

Three equations, one line?

A straight line can be written in more than one form. Gradient-intercept, point-gradient and standard form describe the same set of points, and each is quickest for a different starting fact.

Number and AlgebraNo calculator

Warm-up

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

  1. From Year 8: solve 3x + 2 = 14.
    Answerx = 4.
  2. From Year 9: what is the gradient of the line through (0, 1) and (2, 7)?
    Answer3, because 6 ÷ 2 = 3.
  3. From Year 9: a line has equation y = 2x − 3. What is y when x = 5?
    Answer7, because 10 − 3 = 7.
Words used in this lesson
intercept
the point where a line crosses an axis. For example: (0, c).
gradient-intercept form
y = mx + c, showing gradient and y-intercept. For example: y = 3x − 1.
point-gradient form
y − y₁ = m(x − x₁), from a point and a gradient. For example: y − 2 = 3(x − 1).
standard form
ax + by + d = 0 with whole numbers. For example: 2x − y = 0.
substitute
replace a letter with a value. For example: x = 2.
linear model
a straight-line rule for a real situation. For example: cost = 50 plus 30 per hour.

Same line, or different?

A taxi company in Nairobi publishes its fare rule four different ways on four signs:

Predict first: which signs describe the same fare rule? Test a point or two on each, then use the figure to see one line written three ways.

1
2
3
y = x/2 + 3gradient-intercept form
x − 2y + 6 = 0standard form

Gradient 1/2 and y-intercept 3: y = x/2 + 3. It passes through (2, 4), so y − 4 = (1/2)(x − 2). In standard form with whole numbers this is x − 2y + 6 = 0.

Show me the steps
  1. Rise 1, run 2, c = 3: y = x/2 + 3. Through (2, 4): y − 4 = (1/2)(x − 2). Standard form: x − 2y + 6 = 0.
  2. Rise 3, run 1, c = −2: y = 3x − 2. Through (1, 1): y − 1 = 3(x − 1). Standard form: 3x − y − 2 = 0.
  3. Rise 0, c = 4: the line is horizontal, y = 4. In standard form it is y − 4 = 0.
  4. Rise −2, run 4, c = 1: the gradient is −1/2 and the line is y = −x/2 + 1. In standard form it is x + 2y − 2 = 0.

Three forms of one line. Gradient-intercept: y = mx + c, quick when you know the gradient and where it crosses the y-axis. Point-gradient: y − y₁ = m(x − x₁), quick when you know the gradient and any one point. Standard form: ax + by + d = 0 with whole numbers, no common factor and a positive first coefficient, which can describe any line.

From two points: find m = rise ÷ run, then put one point into the form you want. Parallel lines share m; perpendicular lines have m₁ × m₂ = −1.

Two common mistakes. Sign slips with c: a line through (2, 5) with gradient 3 has 5 = 3 × 2 + c, so c = −1, not 1. And leaving a fraction in standard form: x/2 + y = 3 becomes x + 2y − 6 = 0 after multiplying by 2.

Worked examples

1. Find the equation of the line through (1, 5) and (4, 11).

  1. m = (11 − 5) ÷ (4 − 1) = 6 ÷ 3 = 2. So y = 2x + c.
  2. Use (1, 5): 5 = 2 × 1 + c, so c = 3. The line is y = 2x + 3.

2. Write y = (2/3)x − 4 in standard form with whole numbers.

  1. Multiply every term by 3: 3y = 2x − 12.
  2. Move everything to one side: 2x − 3y − 12 = 0.

3. Find the line parallel to y = −3x + 1 that passes through (2, 1).

  1. Parallel lines share a gradient, so m = −3. Point-gradient form: y − 1 = −3(x − 2).
  2. Expand: y − 1 = −3x + 6, so y = −3x + 7.

Your go. The first step is given. Find the equation of the line through (0, 5) and (2, 1) in the form y = mx + c. The gradient is (1 − 5) ÷ (2 − 0) = −2.

  1. The gradient is −2, so y = −2x + c.
  2. Use the point (0, 5) to find c.
Check the answer

At x = 0, y = 5, so c = 5 and the line is y = −2x + 5. Check with (2, 1): −4 + 5 = 1.

Check before you go on

1. Which is y = 2x − 6 written in standard form with whole numbers?

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. A line has gradient 3 and passes through (0, −4). It has equation y = 3x + c. What is c?

3. The line y = 5x + c passes through (2, 7). What is c?

Level 2 · Building

4. The line through (−1, 4) and (2, 13) has equation y = mx + c. What is c?

5. The line 2x + 4y − 12 = 0 is rearranged to y = mx + c. What is m, as a decimal?

6. A line is parallel to y = 4x − 1 and passes through (0, 5). What is its y value when x = 3?

Level 3 · Stretch

7. A line is perpendicular to y = 2x + 1 and passes through (4, 3). It has equation y = mx + c. What is c?

8. A taxi fare in Lagos is linear in the distance. A 4 km trip costs 1400 naira and a 10 km trip costs 2900 naira. What is the fare for a 15 km trip, in naira?

Something to think about

The equations 2x − y + 4 = 0 and 4x − 2y + 8 = 0 look different. Do they describe the same line? Why might we agree a convention for standard form?

One way to think about it

Yes: the second is the first multiplied by 2, so every point on one is on the other. Without a convention there are infinitely many standard forms for one line (multiply by any non-zero number), so we ask for whole numbers with no common factor and a positive first coefficient. Then each line has one standard form.

Exit ticket

9. Which equation is the line through (1, 7) and (3, 3)?