Steply

Point-slope form

Algebra 1 · 22 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Find the slope
  2. Rise = m × run
  3. Write as y = mx + b
  4. Check

The steps for Point-slope (1, 2) (3, 8). Every solution names its steps, so you always know what comes next.

Worked examples

Line through (2, 3) with slope 4 Easy

The question

Question: write the line through (2, 3) with slope 4 in point-slope form.

(2, 3) and m = 4

1 Rise = m × run

We call the slope m.

m = rise over run.

So the rise is m times the run.

rise = m × run

Like 8/2 is 4, so 8 is 4 times 2.

82 = 4 → 8 = 4 × 2

Take any point (x, y) on the line.

We go from (2, 3) to (x, y).

The rise is y − 3: the change in y.

The run is x − 2: the change in x.

rise = y − 3
run = x − 2

Put them in: rise = m × run becomes y − 3 = 4(x − 2).

y − 3 = 4(x − 2)

This is called point-slope form.

It is made from a point and the slope.

y − y1 = m(x − x1)

2 Write as y = mx + b

The 4 in front of the bracket multiplies every part inside.

So we write 4 times each part.

y − 3 = 4 × x + 4 × (−2)

Work out each multiplication.

Now we have y − 3 = 4x − 8.

y − 3 = 4x − 8

We want y by itself.

The − 3 is in the way.

To remove − 3, we add 3.

We do it on both sides, so they stay equal.

y − 3 + 3 = 4x − 8 + 3

− 3 and + 3 make zero.

Cross them out.

−8 plus 3 is −5.

So y = 4x − 5.

y = 4x − 5

3 Check

Let's check: put x = 2 into y = 4x − 5.

We get 3, the y of (2, 3).

y = 4(2) − 5 = 8 − 5 = 3

And the number in front of x is 4: the slope.

Right!

m = 4

Let's look back.

First: rise = m × run.

Then: write as y = mx + b.

Last: check.

Answer
y − 3 = 4(x − 2)

▶ Watch it solved

Line through (-1, 4) with slope -2 Medium

The question

Question: write the line through (−1, 4) with slope −2 in point-slope form.

(−1, 4) and m = −2

1 Rise = m × run

We call the slope m.

m = rise over run.

So the rise is m times the run.

rise = m × run

Like 8/2 is 4, so 8 is 4 times 2.

82 = 4 → 8 = 4 × 2

Take any point (x, y) on the line.

We go from (−1, 4) to (x, y).

The rise is y − 4: the change in y.

The run is x − (−1): the change in x.

rise = y − 4
run = x − (−1)

Put them in: rise = m × run becomes y − 4 = −2(x − (−1)).

y − 4 = −2(x − (−1))

Taking away a negative is the same as adding.

− (−) = +

So it becomes y − 4 = −2(x + 1).

y − 4 = −2(x + 1)

This is called point-slope form.

It is made from a point and the slope.

y − y1 = m(x − x1)

2 Write as y = mx + b

The −2 in front of the bracket multiplies every part inside.

So we write −2 times each part.

y − 4 = −2 × x − 2 × 1

Work out each multiplication.

Now we have y − 4 = −2x − 2.

y − 4 = −2x − 2

We want y by itself.

The − 4 is in the way.

To remove − 4, we add 4.

We do it on both sides, so they stay equal.

y − 4 + 4 = −2x − 2 + 4

− 4 and + 4 make zero.

Cross them out.

−2 plus 4 is 2.

So y = −2x + 2.

y = −2x + 2

3 Check

Let's check: put x = −1 into y = −2x + 2.

We get 4, the y of (−1, 4).

y = −2(−1) + 2 = 2 + 2 = 4

And the number in front of x is −2: the slope.

Right!

m = −2

Let's look back.

First: rise = m × run.

Then: write as y = mx + b.

Last: check.

Answer
y − 4 = −2(x + 1)

▶ Watch it solved

Point-slope (1, 2) (3, 8) Harder

The question

Question: write the line through (1, 2) and (3, 8) in point-slope form.

(1, 2) and (3, 8)

1 Find the slope

We call the slope m.

m = rise over run.

The top is the change in y: second y minus first y.

The bottom is the change in x: second x minus first x.

m = riserun = y2 − y1x2 − x1

Put the numbers in.

The second point is (3, 8).

The first point is (1, 2).

m = 8 − 23 − 1

8 minus 2 is 6.

3 minus 1 is 2.

m = 62

6 divided by 2 is 3.

So m = 3.

m = 3

2 Rise = m × run

m = rise over run.

So the rise is m times the run.

rise = m × run

Like 8/2 is 4, so 8 is 4 times 2.

82 = 4 → 8 = 4 × 2

Take any point (x, y) on the line.

We go from (1, 2) to (x, y).

The rise is y − 2: the change in y.

The run is x − 1: the change in x.

rise = y − 2
run = x − 1

Put them in: rise = m × run becomes y − 2 = 3(x − 1).

y − 2 = 3(x − 1)

This is called point-slope form.

It is made from a point and the slope.

y − y1 = m(x − x1)

3 Write as y = mx + b

The 3 in front of the bracket multiplies every part inside.

So we write 3 times each part.

y − 2 = 3 × x + 3 × (−1)

Work out each multiplication.

Now we have y − 2 = 3x − 3.

y − 2 = 3x − 3

We want y by itself.

The − 2 is in the way.

To remove − 2, we add 2.

We do it on both sides, so they stay equal.

y − 2 + 2 = 3x − 3 + 2

− 2 and + 2 make zero.

Cross them out.

−3 plus 2 is −1.

So y = 3x − 1.

y = 3x − 1

4 Check

Let's check: put x = 1 into y = 3x − 1.

We get 2, the y of (1, 2).

y = 3(1) − 1 = 3 − 1 = 2

Now x = 3 into y = 3x − 1.

We get 8, the y of (3, 8).

y = 3(3) − 1 = 9 − 1 = 8

And the number in front of x is 3: the slope.

Right!

m = 3

Let's look back.

First: find the slope.

Then: rise = m × run.

Then: write as y = mx + b.

Last: check.

Answer
y − 2 = 3(x − 1)

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. line through (2, 3) with slope 4
    Answer
    y − 3 = 4(x − 2)
  2. line through (1, 5) with slope 2
    Answer
    y − 5 = 2(x − 1)
  3. line through (3, 2) with slope 3
    Answer
    y − 2 = 3(x − 3)
  4. line through (4, 1) with slope 2
    Answer
    y − 1 = 2(x − 4)
  5. line through (1, 1) with slope 5
    Answer
    y − 1 = 5(x − 1)
  6. line through (2, 6) with slope 1
    Answer
    y − 6 = (x − 2)
  7. line through (5, 4) with slope 3
    Answer
    y − 4 = 3(x − 5)
  8. line through (3, 7) with slope 2
    Answer
    y − 7 = 2(x − 3)

Medium

  1. line through (-1, 4) with slope -2
    Answer
    y − 4 = −2(x + 1)
  2. line through (2, -3) with slope 4
    Answer
    y + 3 = 4(x − 2)
  3. line through (-2, -5) with slope 3
    Answer
    y + 5 = 3(x + 2)
  4. line through (2, 2) with slope -1
    Answer
    y − 2 = −(x − 2)
  5. line through (4, 0) with slope 1/2
    Answer
    y = 12(x − 4)
  6. line through (0, 5) with slope 3
    Answer
    y − 5 = 3x
  7. line through (6, 1) with slope -2/3
    Answer
    y − 1 = −23(x − 6)
  8. line through (-3, 2) with slope -1
    Answer
    y − 2 = −(x + 3)
  9. line through (2, -3) with slope -3/2
    Answer
    y + 3 = −32(x − 2)

Harder

  1. point-slope (1, 2) (3, 8)
    Answer
    y − 2 = 3(x − 1)
  2. point-slope (-1, 5) (2, -1)
    Answer
    y − 5 = −2(x + 1)
  3. point-slope (2, 1) (6, 3)
    Answer
    y − 1 = 12(x − 2)
  4. point-slope (-2, -1) (1, 5)
    Answer
    y + 1 = 2(x + 2)
  5. point-slope (0, -2) (3, 4)
    Answer
    y + 2 = 2x