Point-Slope Form

Point-slope form is a useful way to write the equation of a straight line when you know:

• one point on the line
• the slope of the line

Instead of first finding the y-intercept, you can use the known point and slope directly.

The general form is:

y − y₁ = m(x − x₁)

where:

• m = slope
• (x₁, y₁) = a known point on the line
• (x, y) = any other point on the line

Point-slope form is especially useful when the known point is not the y-intercept.

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Understanding Point-Slope Form

Consider a line with:

slope = 2

and a known point:

(3, 5)

Point-slope form allows us to immediately write:

y − 5 = 2(x − 3)

Notice how each piece of information appears directly in the equation:

m = 2

x₁ = 3

y₁ = 5

Therefore:

y − y₁ = m(x − x₁)

becomes:

y − 5 = 2(x − 3)

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Why Point-Slope Form Works

Slope is defined as:

m = change in y/change in x

Using a known point (x₁, y₁) and another point (x, y):

m = (y − y₁)/(x − x₁)

Multiply both sides by:

(x − x₁)

This gives:

y − y₁ = m(x − x₁)

So point-slope form comes directly from the definition of slope.

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When Point-Slope Form Is Useful

Point-slope form is particularly useful when a problem gives:

a slope and one point

For example:

Write the equation of a line with slope 4 that passes through (2, 7).

There is no need to find the y-intercept first.

Simply substitute:

m = 4

x₁ = 2

y₁ = 7

Therefore:

y − 7 = 4(x − 2)

This is already a complete equation of the line.

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Point-Slope Form vs Slope-Intercept Form

Two common forms of a linear equation are:

Point-slope form

y − y₁ = m(x − x₁)

Slope-intercept form

y = mx + b

Point-slope form is usually most convenient when you know:

a point and a slope

Slope-intercept form is usually most convenient when you know:

the slope and y-intercept

Both forms can describe the same line.

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Identifying the Known Point

Consider:

y − 6 = 3(x − 2)

Compare with:

y − y₁ = m(x − x₁)

We can identify:

m = 3

x₁ = 2

y₁ = 6

Therefore, the line:

• has slope 3
• passes through (2, 6)

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Negative Coordinates Require Care

Consider:

y + 4 = 2(x − 3)

Point-slope form is:

y − y₁ = m(x − x₁)

Notice:

y + 4 = y − (−4)

Therefore:

y₁ = −4

The known point is:

(3, −4)

and:

m = 2

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Two Negative Coordinates

Consider:

y + 5 = −3(x + 2)

Rewrite mentally as:

y − (−5) = −3[x − (−2)]

Therefore:

m = −3

and the point is:

(−2, −5)

This is one of the most common places students make sign errors.

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Writing Point-Slope Form from a Point and Slope

Suppose:

m = 5

point = (4, 7)

Start with:

y − y₁ = m(x − x₁)

Substitute:

y − 7 = 5(x − 4)

That is the equation in point-slope form.

There is no need to simplify unless the question asks for another form.

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Worked Example: Positive Slope

Write the equation of the line with:

slope = 3

passing through:

(2, 8)

Substitute:

m = 3

x₁ = 2

y₁ = 8

Therefore:

y − 8 = 3(x − 2)

This is the point-slope equation.

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Worked Example: Negative Slope

Write the equation of a line with:

m = −2

passing through:

(4, 5)

Substitute:

y − 5 = −2(x − 4)

The negative slope means the line falls from left to right.

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Worked Example: Negative Coordinate

Write the equation of a line with:

m = 4

passing through:

(−3, 2)

Substitute:

y − 2 = 4[x − (−3)]

Subtracting a negative becomes addition:

y − 2 = 4(x + 3)

This is the correct point-slope equation.

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Worked Example: Both Coordinates Negative

Write the equation of a line with:

m = −1/2

passing through:

(−4, −6)

Substitute:

y − (−6) = −1/2[x − (−4)]

Simplify:

y + 6 = −1/2(x + 4)

The signs come directly from subtracting the coordinates.

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Point-Slope Form with Zero Slope

Suppose a line has:

m = 0

and passes through:

(3, 7)

Point-slope form gives:

y − 7 = 0(x − 3)

Therefore:

y − 7 = 0

and:

y = 7

This is a horizontal line.

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Vertical Lines

A vertical line has:

undefined slope

Therefore, the standard point-slope form cannot be used with a numerical value of m for a vertical line.

If a vertical line passes through:

(4, 7)

every point on the line has:

x = 4

Therefore, its equation is:

x = 4

This is an important exception.

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Graphing from Point-Slope Information

Suppose a line has:

m = 2/3

and passes through:

(1, 2)

Plot the known point:

(1, 2)

Then use:

m = 2/3

which means:

right 3

up 2

This gives another point:

(4, 4)

Repeat the same movement if needed and draw the line.

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Converting to Slope-Intercept Form

Point-slope form can be converted into:

y = mx + b

The main steps are:

1. distribute the slope
2. simplify
3. isolate y

For example:

y − 5 = 2(x − 3)

Distribute:

y − 5 = 2x − 6

Add 5 to both sides:

y = 2x − 1

Therefore:

point-slope form:

y − 5 = 2(x − 3)

slope-intercept form:

y = 2x − 1

Both equations describe exactly the same line.

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Worked Conversion Example

Convert:

y − 4 = 3(x + 2)

to slope-intercept form.

Distribute:

y − 4 = 3x + 6

Add 4:

y = 3x + 10

Therefore:

m = 3

b = 10

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Conversion with a Negative Slope

Convert:

y − 6 = −2(x − 5)

Distribute:

y − 6 = −2x + 10

Add 6:

y = −2x + 16

The slope remains:

−2

Converting the form of an equation does not change the line.

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Conversion with Negative Coordinates

Convert:

y + 3 = 4(x + 2)

Distribute:

y + 3 = 4x + 8

Subtract 3:

y = 4x + 5

The original point was:

(−2, −3)

Check:

−3 = 4(−2) + 5

−3 = −8 + 5

−3 = −3

Correct.

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Finding the y-Intercept from Point-Slope Form

Sometimes converting to slope-intercept form reveals useful information.

Consider:

y − 7 = 3(x − 2)

Expand:

y − 7 = 3x − 6

Add 7:

y = 3x + 1

Therefore:

m = 3

b = 1

The y-intercept is:

(0, 1)

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Finding the Equation from Two Points

Sometimes the slope is not given directly.

Suppose a line passes through:

(2, 3) and (6, 11)

First calculate the slope:

m = (11 − 3)/(6 − 2)

m = 8/4

m = 2

Now choose either known point.

Using (2, 3):

y − 3 = 2(x − 2)

This is the point-slope equation.

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Either Point Can Be Used

Using the other point:

(6, 11)

we get:

y − 11 = 2(x − 6)

These equations look different:

y − 3 = 2(x − 2)

y − 11 = 2(x − 6)

but they describe:

the same line

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Verifying That the Equations Are Equivalent

First equation:

y − 3 = 2(x − 2)

Expand:

y − 3 = 2x − 4

y = 2x − 1

Second equation:

y − 11 = 2(x − 6)

Expand:

y − 11 = 2x − 12

y = 2x − 1

Both simplify to the same slope-intercept equation.

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Determining Whether a Point Lies on a Line

To determine whether a point lies on a line:

1. substitute its x-coordinate for x
2. substitute its y-coordinate for y
3. simplify both sides
4. check whether the equation is true

If the equation is true:

the point lies on the line

If it is false:

the point does not lie on the line

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Worked Example: Point on the Line

Does:

(4, 9)

lie on:

y − 3 = 2(x − 1)?

Substitute:

9 − 3 = 2(4 − 1)

6 = 2(3)

6 = 6

The statement is true.

Therefore:

(4, 9) lies on the line.

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Worked Example: Point Not on the Line

Does:

(5, 8)

lie on:

y − 2 = 3(x − 1)?

Substitute:

8 − 2 = 3(5 − 1)

6 = 12

This is false.

Therefore:

(5, 8) does not lie on the line.

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Why Substitution Works

An equation of a line describes:

every point on that line

Any coordinate pair on the line must make the equation true.

A coordinate pair not on the line will not satisfy the equation.

This idea applies to equations far beyond point-slope form.

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Checking the Given Point

The point used to create a point-slope equation should always satisfy the equation automatically.

For:

y − 5 = 3(x − 2)

the given point is:

(2, 5)

Substitute:

5 − 5 = 3(2 − 2)

0 = 3(0)

0 = 0

Correct.

This provides a quick check for sign errors.

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Writing an Equation from a Graph

Suppose a graph clearly shows:

a point at (3, 4)

and slope:

−2

Point-slope form gives:

y − 4 = −2(x − 3)

If slope-intercept form is required:

y − 4 = −2x + 6

y = −2x + 10

Point-slope form can therefore be an efficient intermediate step when working from graphs.

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Why Point-Slope Form Can Be Faster

Suppose you know:

m = 4

and:

(7, 12)

Using slope-intercept form immediately would require:

12 = 4(7) + b

12 = 28 + b

b = −16

then:

y = 4x − 16

Point-slope form allows:

y − 12 = 4(x − 7)

immediately.

Both are correct, but point-slope form requires fewer initial steps.

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Real-World Meaning

Point-slope form is useful when you know:

• a rate of change
• one known measurement

For example, suppose the temperature increases steadily at:

2°C per hour

and after 3 hours the temperature is:

18°C

Let:

x = time in hours

y = temperature in °C

Known point:

(3, 18)

Slope:

2°C/hour

Therefore:

y − 18 = 2(x − 3)

This equation models the temperature.

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Converting the Temperature Model

Start with:

y − 18 = 2(x − 3)

Expand:

y − 18 = 2x − 6

Add 18:

y = 2x + 12

This tells us that the model predicts:

12°C at x = 0

Point-slope form emphasized the known observation.

Slope-intercept form reveals the predicted starting value.

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Real-World Example: Savings

Suppose a person saves:

$30 per week

After 5 weeks, they have:

$250

Let:

x = weeks

y = savings in dollars

Slope:

m = 30

Known point:

(5, 250)

Point-slope equation:

y − 250 = 30(x − 5)

Convert:

y − 250 = 30x − 150

y = 30x + 100

The model indicates an initial amount of:

$100

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Real-World Example: Water Level

A reservoir's water level is decreasing at:

0.4 m per day

On day 10, its measured level is:

18 m

Slope:

m = −0.4

Known point:

(10, 18)

Point-slope model:

y − 18 = −0.4(x − 10)

This directly incorporates the measured value and the rate of change.

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Using the Model to Make a Prediction

Using:

y − 18 = −0.4(x − 10)

find the predicted level on day 15.

Substitute:

x = 15

y − 18 = −0.4(15 − 10)

y − 18 = −0.4(5)

y − 18 = −2

y = 16

The predicted water level is:

16 m

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Finding an Unknown Point

Suppose a line has slope:

3

and passes through:

(2, 5)

Find y when:

x = 6

Write:

y − 5 = 3(x − 2)

Substitute x = 6:

y − 5 = 3(6 − 2)

y − 5 = 12

y = 17

Therefore, the point:

(6, 17)

lies on the line.

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Finding an Unknown x-Coordinate

Suppose:

y − 4 = 2(x − 3)

and a point on the line has:

y = 14

Substitute:

14 − 4 = 2(x − 3)

10 = 2(x − 3)

5 = x − 3

x = 8

Therefore:

(8, 14)

lies on the line.

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Comparing Lines

Suppose:

Line A:

y − 3 = 2(x − 1)

Line B:

y + 5 = 2(x − 4)

Both have:

m = 2

Therefore, the lines have the same slope.

Convert them if needed.

Line A:

y − 3 = 2x − 2

y = 2x + 1

Line B:

y + 5 = 2x − 8

y = 2x − 13

They have different y-intercepts.

Therefore, they are:

parallel

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Lines Through the Same Point

Consider:

y − 4 = 2(x − 3)

and:

y − 4 = −3(x − 3)

Both lines pass through:

(3, 4)

But their slopes are different.

Therefore, they intersect at:

(3, 4)

Point-slope form makes this relationship particularly easy to see.

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Connecting Point-Slope and Slope-Intercept Forms

Consider:

y − y₁ = m(x − x₁)

Expand:

y − y₁ = mx − mx₁

Add y₁:

y = mx + (y₁ − mx₁)

Compare with:

y = mx + b

Therefore:

b = y₁ − mx₁

This shows mathematically that point-slope form and slope-intercept form describe the same family of nonvertical straight lines.

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Choosing the Best Form

Different forms are useful in different situations.

There is no requirement to use only one form forever.

Strong algebra students choose the form that makes the problem easiest.

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Point-Slope Form and Parallel Lines

Suppose you need a line parallel to:

y = 3x + 7

that passes through:

(4, −2)

Parallel lines have the same slope.

Therefore:

m = 3

Use the new point:

(4, −2)

Point-slope form:

y − (−2) = 3(x − 4)

Therefore:

y + 2 = 3(x − 4)

This is a convenient way to construct a parallel line.

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Solving a Context Problem

A delivery company knows that the total distance travelled increases at a constant rate of:

60 km/h

After 2 hours, the vehicle has travelled:

135 km

Let:

x = time in hours

y = distance in kilometres

Known point:

(2, 135)

Slope:

60

Equation:

y − 135 = 60(x − 2)

To find the predicted distance at 5 hours:

y − 135 = 60(5 − 2)

y − 135 = 180

y = 315 km

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Interpreting the Result Carefully

A mathematical model may produce values outside the realistic situation.

For example, extending a constant-speed model backward could produce a mathematically valid y-intercept that does not represent an actual measured starting condition.

Therefore, distinguish between:

what the equation predicts

and

what is meaningful in the real-world context

Always consider the relevant domain.

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A Reliable Strategy for Writing Point-Slope Form

When given a point and slope:

1. Write y − y₁ = m(x − x₁).
2. Identify m.
3. Identify x₁.
4. Identify y₁.
5. Substitute the values carefully.
6. Keep parentheses around negative coordinates.
7. Simplify double negatives.
8. Check that the known point satisfies the equation.
9. Leave the equation in point-slope form unless another form is requested.

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A Reliable Strategy for Converting Forms

To convert point-slope form to slope-intercept form:

1. Start with the point-slope equation.
2. Distribute m across the parentheses.
3. Simplify the right side.
4. Add or subtract the appropriate value to isolate y.
5. Write the final equation as y = mx + b.
6. Check that the slope has not changed.
7. Identify the y-intercept.
8. Substitute the original point to verify the equation.

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A Reliable Strategy for Testing a Point

To determine whether a point lies on a line:

1. Identify the point's x-coordinate.
2. Identify its y-coordinate.
3. Substitute both values into the equation.
4. Simplify the left side.
5. Simplify the right side.
6. Compare the results.
7. If both sides are equal, the point lies on the line.
8. If they are not equal, the point does not lie on the line.

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Common Mistakes

Using the wrong formula

Point-slope form is:

y − y₁ = m(x − x₁)

Switching x and y coordinates

For (3, 7):

x₁ = 3

y₁ = 7

not the other way around.

Forgetting that subtracting a negative becomes addition

For the point (−4, −2):

y − (−2) = m[x − (−4)]

becomes:

y + 2 = m(x + 4)

Changing the slope during conversion

Converting forms changes the appearance of the equation, not the line itself.

Thinking point-slope form requires the y-intercept

It requires any known point on the line.

Assuming the point in the equation is always positive

The point can contain positive, negative, or zero coordinates.

Testing only the x-coordinate

A point consists of both x and y. Both values must satisfy the equation.

Assuming a vertical line has slope zero

Vertical lines have undefined slope and require equations such as x = 4.

Finding b before using point-slope form

You can do this, but it defeats the main advantage of point-slope form when a point and slope are already known.

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Key Terms

Point-slope form — linear equation written as y − y₁ = m(x − x₁).

Slope (m) — constant rate of change and measure of a line's steepness and direction.

Known point — coordinate pair (x₁, y₁) known to lie on the line.

Coordinate pair — ordered pair (x, y) representing a point.

Slope-intercept form — linear equation written as y = mx + b.

y-intercept — point where a line crosses the y-axis.

Linear equation — equation whose graph is a straight line.

Rate of change — change in one variable compared with change in another.

Substitution — replacing a variable with a known value.

Equivalent equations — equations written differently that describe the same relationship.

Parallel lines — distinct coplanar lines with equal slopes that never intersect.

Undefined slope — slope of a vertical line.

Domain — set of allowable input values for a mathematical relationship or model.

Linear model — straight-line relationship used to represent a constant rate of change.

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Key Takeaways

• Point-slope form is y − y₁ = m(x − x₁).
• It is especially useful when one point and the slope are known.
• The point (x₁, y₁) can be any point on the line.
• The known point does not need to be the y-intercept.
• The value m represents the slope.
• Negative coordinates require careful handling because subtracting a negative becomes addition.
• A point and slope determine exactly one nonvertical line.
• Two points can be used to find the slope before writing point-slope form.
• Either of the two known points can be used to write the equation.
• Different-looking point-slope equations can represent the same line.
• Point-slope form can be converted to slope-intercept form by distributing and isolating y.
• Converting forms does not change the slope or graph.
• Substitution can determine whether a point lies on a line.
• A point lies on a line only if its coordinates make the equation true.
• Point-slope form is useful in real-world situations involving a known measurement and a constant rate of change.
• Point-slope form is often convenient for constructing parallel lines because the required slope can be inserted directly.
• Vertical lines cannot be written using ordinary point-slope form because their slope is undefined.
• Choosing the most convenient equation form can make linear problems much easier to solve.

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Check Your Understanding

1. Write the general equation for point-slope form and explain what m, x₁, and y₁ represent.
2. Write the equation of a line with slope 4 that passes through (3, 7).
3. Write the equation of a line with slope −2 that passes through (5, 1).
4. Write the point-slope equation of a line with slope 3/4 passing through (−2, 5).
5. A line is written as y + 3 = 5(x − 4). Identify its slope and the point used to write the equation.
6. A line is written as y − 6 = −2(x + 3). Identify the slope and the known point.
7. Convert y − 4 = 3(x − 2) into slope-intercept form.
8. Convert y + 5 = −2(x + 1) into slope-intercept form.
9. Determine whether (4, 11) lies on the line y − 3 = 2(x − 1). Show your substitution.
10. Determine whether (5, 7) lies on y + 1 = 3(x − 2).
11. Find the equation of the line passing through (2, 5) and (6, 13). First calculate the slope, then write the equation in point-slope form.
12. A line has slope −3 and passes through (4, 10). Find the value of y when x = 7.
13. A person's savings increase by $25 per week. After 6 weeks, the person has $280. Write a point-slope equation for the savings and then convert it to slope-intercept form.
14. Write an equation for a line parallel to y = 4x − 7 that passes through (3, 2).
15. A student is asked to write the equation of a line with slope 2 passing through (−3, −5) and writes y − 5 = 2(x − 3). Explain both sign errors and write the correct point-slope equation.