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G4 · Convert a line equation to slope-intercept form

Learn to convert a line equation to slope-intercept form through clear examples and targeted practice.

Ontario Grade 10 Mathematics

Modelling Linear Relations

Use balanced equation steps to isolate y and read the slope and y-intercept

A line can be described by an equation in more than one way. One equation may show both variables on the same side. Slope-intercept form makes the value of y clear and makes two features of the line easy to read. You will use familiar equation-solving steps: keep both sides balanced, use inverse operations, and isolate y. This conversion works when the equation can be solved for y. For an equation written as Ax + By = C, this requires B to be nonzero.

What you will learn

Recognize slope-intercept form and identify its slope and y-intercept.
Use inverse operations to rearrange a line equation into slope-intercept form when y can be isolated.
Check a rearranged equation by comparing it with the original.

1. Grade 9 bridge: keep an equation balanced

An equation says that two expressions have the same value. Think of the equals sign as a balance. If you add, subtract, multiply, or divide on one side, do the same to the other side to keep the equation true.

Inverse operations undo one another. Addition and subtraction are inverse operations. Multiplication and division are also inverse operations. To isolate a variable means to leave it by itself on one side of an equation.

For example, if a number has 5 added to it, subtracting 5 undoes the addition. The same idea works when a line equation contains x and y. The goal is to rearrange the equation without changing which pairs of x and y make it true.

Do the same operation to both sides of an equation.
Use inverse operations to isolate the variable you want.

2. Recognize slope-intercept form

Slope-intercept form is a way to write a linear equation. A linear equation has a straight-line graph. The pattern for slope-intercept form is y = mx + b. Here, m and b stand for numbers, not extra variables that must have particular values.

The number m is the slope. It tells how much y changes when x increases by 1. The number b is the y-intercept: the value of y where the line crosses the vertical axis. At that crossing, x is 0.

For example, in y = 2x + 3, the slope is 2 and the y-intercept is 3. The important feature of the form is that y is by itself and the right side has an x term and a constant term.

An equation may be written as Ax + By = C, where A, B, and C are fixed numbers. It can be converted to slope-intercept form when it can be solved for y. In this type of equation, that is possible when B is not zero, because isolating y requires dividing by B. If B is zero, the equation has no y term to isolate, so it cannot be rearranged into y = mx + b. In the conversion steps, keep track of negative signs and divide every term on the other side when needed.

y=mx+b

Slope-intercept form has the pattern y = mx + b.
The coefficient of x is the slope; the constant term is the y-intercept.
The conversion applies when y can be isolated. In Ax + By = C, B must be nonzero.

3. Rearrange one step at a time

Suppose an equation has an x term and a y term on the left. First, undo the addition or subtraction involving x. Apply that operation to both sides. This leaves the y term on one side.

Next, undo the number multiplying y. Divide both sides by that number. Divide every term on the other side as well. For example, dividing 8 − 3x by 2 gives 4 − 3x/2, not 8 − 3x/2.

A final equation may have its constant written before its x term. Reorder the terms so the x term comes first, as in y = mx + b. Reordering the terms in a sum does not change its value.

The table tracks a conversion. Each row keeps the equation balanced and moves closer to having y alone. The values that make the starting equation true also make the rearranged equation true.

Undo the x term first, using the same operation on both sides.
Divide every term when isolating y.
Write the x term before the constant in the final form.

4. Independent practice and a final check

Try these conversions on your own. Show each balanced step, then identify the slope and y-intercept from the final equation.

Practice A: Convert x + y = 6. Practice B: Convert 4x + 2y = 10. Practice C: Convert 3x − y = 2. In Practice C, notice that the coefficient of y is negative, so dividing by it changes the signs.

To check a conversion, choose a value of x and find the matching y in both the original and rearranged equations. If the equations are equivalent, the results agree. You can also check that the final equation has y alone and matches the pattern y = mx + b.

A negative coefficient of y means dividing by a negative number.
Read the slope and y-intercept only after reaching slope-intercept form.
Substitution into both equations can confirm that a rearrangement is correct.

A balanced rearrangement

Equation	Operation used
3x + 2y = 8	Starting equation
2y = 8 − 3x	Subtract 3x from both sides
y = 4 − 3x/2	Divide both sides by 2
y = −3x/2 + 4	Write the x term first

Worked example

Convert a line equation with a coefficient of y

Rewrite 3x + 2y = 8 in slope-intercept form. Then state its slope and y-intercept.

Move the x term
The term 3x is added to 2y. Subtract 3x from both sides. This leaves the y term on one side, and doing the same subtraction on each side keeps the equation balanced.
$2y=8-3x$
Isolate y
The variable y is multiplied by 2. Divide both sides by 2. Divide each term on the right by 2 so the equation stays balanced.
$y=4-\frac{3}{2}x$
Write the x term first
The expression 4 − 3x/2 has the same value as −3x/2 + 4. Writing the x term first makes the slope and constant term easy to identify.
$y=-\frac{3}{2}x+4$
Read the slope and y-intercept
Compare the final equation with y = mx + b. The coefficient of x is m, the slope. The constant term is b, the y-intercept.
$m=-\frac{3}{2},\quad b=4$

Answer: The slope-intercept form is y = −3x/2 + 4. The slope is −3/2, and the y-intercept is 4.

Check: When x = 0, the original equation becomes 2y = 8, so y = 4. The rearranged equation also gives y = 4 when x = 0. This confirms the y-intercept.

Common mistakes and how to avoid them

Subtracting 3x on the left but not on the right.

Correction: Apply the same operation to both sides. Otherwise, the equation may no longer be balanced.

Dividing only the first term on the right by the coefficient of y.

Correction: Divide every term on the right. For example, dividing 8 − 3x by 2 gives 4 − 3x/2.

Calling the constant term the slope.

Correction: In y = mx + b, the slope is the coefficient of x, and the y-intercept is the constant term.

Assuming every equation written as Ax + By = C can be put in slope-intercept form.

Correction: The conversion requires y to be isolatable. In Ax + By = C, B must be nonzero; if B is zero, there is no y term to isolate.

Lesson summary

Slope-intercept form is y = mx + b.
Use inverse operations on both sides to isolate y.
Divide every term when dividing an expression with multiple terms.
In the final form, m is the slope and b is the y-intercept.
For Ax + By = C, the conversion requires B to be nonzero.

Check your understanding

Question 1

What is the slope-intercept form of 2x + y = 7?

y = −2x + 7
y = 2x + 7
y = −2x − 7
y = 7x − 2

Show answer and explanation

y = −2x + 7

Subtract 2x from both sides to get y = 7 − 2x. Write the x term first to get y = −2x + 7.

Question 2

What is the slope in y = 5x − 3?

−3
5
−5
3

Show answer and explanation

Compare with y = mx + b. The coefficient of x is 5, so the slope is 5.

Question 3

Which equation is in slope-intercept form?

2x + 3y = 12
y = −4x + 1
x − y = 5
3y = 6x + 9

Show answer and explanation

y = −4x + 1

In y = −4x + 1, y is by itself and the equation matches the pattern y = mx + b.

Key terms

Linear equation: An equation whose graph is a straight line.
Slope-intercept form: A way to write a line equation as y = mx + b.
Slope: The amount y changes when x increases by 1; it is the coefficient of x in slope-intercept form.
Y-intercept: The value of y where a line crosses the vertical axis; it is the constant term in slope-intercept form.
Coefficient: A number multiplied by a variable.
Inverse operations: Operations that undo one another, such as addition and subtraction.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic G4. It is a study resource, not an official curriculum publication.

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