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Linear Equations: Definition, Forms, and How to Solve Them

Everything you need for linear equations in Algebra 1: what makes an equation linear, the three forms of a line, step-by-step solving, graphing, and practice problems with answers.

Practice Linear Equations

What is a linear equation?

A linear equation is an equation where each variable appears only to the first power (no x², no √x, no 1/x) and variables are never multiplied together. When you graph it, you always get a straight line, which is where the name comes from.

A linear equation in one variable looks like this, where a, b and c are numbers and a ≠ 0:

ax + b = c

A linear equation in two variables, the kind you graph, looks like this:

y = mx + b

Linear or not? Quick examples

Equation	Linear?	Why
3x − 7 = 11	Yes	One variable, power 1
2x + 4y = 8	Yes	Two variables, each to power 1
y = −½x + 3	Yes	Fractions as coefficients are fine
x = 5	Yes	A vertical line
y = x²	No	x is squared (that's a parabola)
xy = 6	No	Variables multiplied together
y = 1/x	No	Variable in the denominator

The three forms of a linear equation

The same straight line can be written three common ways. Each one makes a different piece of information easy to read.

1. Slope-intercept form

y = mx + b

m is the slope (how steep the line is: rise over run) and b is the y-intercept (where the line crosses the y-axis). In y = 2x + 1, the slope is 2 and the line crosses the y-axis at (0, 1). This is the easiest form to graph.

2. Standard form

Ax + By = C

A, B and C are usually integers, with A not negative. Standard form makes both intercepts easy to find: set y = 0 to get the x-intercept and x = 0 to get the y-intercept. For 2x + 3y = 6, the x-intercept is (3, 0) and the y-intercept is (0, 2).

3. Point-slope form

y − y₁ = m(x − x₁)

Use this when you know one point (x₁, y₁) on the line and its slope m. A line through (1, 4) with slope 3 is y − 4 = 3(x − 1), which simplifies to y = 3x + 1.

Converting standard form to slope-intercept form

Solve for y. Starting from 2x + 3y = 6:

Subtract 2x from both sides: 3y = −2x + 6
Divide every term by 3: y = −⅔x + 2

So the slope is −⅔ and the y-intercept is 2.

How to solve a linear equation, step by step

Solving means finding the value of the variable that makes the equation true. Use the same four steps every time:

Simplify each side. Distribute to remove parentheses and combine like terms.
Collect variables on one side. Add or subtract variable terms so x appears on only one side.
Collect constants on the other side. Add or subtract numbers to move them away from x.
Isolate the variable. Divide (or multiply) both sides by the number in front of x, then check your answer.

The rule underneath all four steps: keep the equation balanced. Whatever you do to one side, you must do to the other.

Worked examples

Example 1: One-step equation

Easy

Solve: x + 7 = 12

Step 1: Subtract 7 from both sides

x = 12 − 7

Undo the +7 with its inverse operation.

Step 2: Simplify

x = 5

Check: 5 + 7 = 12 ✓

Final Answer:

x = 5

Example 2: Two-step equation

Easy

Solve: 3x − 4 = 11

Step 1: Add 4 to both sides

3x = 15

Move the constant away from the variable first.

Step 2: Divide both sides by 3

x = 5

Check: 3(5) − 4 = 11 ✓

Final Answer:

x = 5

Example 3: Variables on both sides

Medium

Solve: 5x + 3 = 2x + 12

Step 1: Subtract 2x from both sides

3x + 3 = 12

Gather the x terms on one side.

Step 2: Subtract 3 from both sides

3x = 9

Now it is a one-step equation.

Step 3: Divide both sides by 3

x = 3

Check: 5(3) + 3 = 18 and 2(3) + 12 = 18 ✓

Final Answer:

x = 3

Example 4: Equation with parentheses

Medium

Solve: 2(x − 3) + 4 = 3x − 5

Step 1: Distribute the 2

2x − 6 + 4 = 3x − 5

Multiply 2 by both terms inside the parentheses.

Step 2: Combine like terms

2x − 2 = 3x − 5

−6 + 4 = −2

Step 3: Subtract 2x from both sides

−2 = x − 5

Keep the x term positive by moving the smaller one.

Step 4: Add 5 to both sides

3 = x

Check: 2(0) + 4 = 4 and 3(3) − 5 = 4 ✓

Final Answer:

x = 3

Example 5: Equation with fractions

Hard

Solve: x/3 + 1/2 = 5/6

Step 1: Multiply every term by the LCD, 6

2x + 3 = 5

Clearing fractions first makes the rest ordinary two-step solving.

Step 2: Subtract 3 from both sides

2x = 2

Step 3: Divide both sides by 2

x = 1

Check: 1/3 + 1/2 = 2/6 + 3/6 = 5/6 ✓

Final Answer:

x = 1

Special cases: no solution and infinitely many solutions

Most linear equations in one variable have exactly one solution, but two special cases come up on tests:

No solution

2(x + 3) = 2x + 5 becomes 2x + 6 = 2x + 5. Subtracting 2x leaves 6 = 5, which is false, so no value of x works.

Infinitely many solutions

3(x − 1) = 3x − 3 becomes 3x − 3 = 3x − 3. Subtracting 3x leaves −3 = −3, which is always true, so every value of x works. This kind of equation is called an identity.

Graphing a linear equation

The graph of a linear equation in two variables is a straight line, so you only need two points to draw it. From slope-intercept form, plot the y-intercept first, then use the slope to find the next point.

y = 2x + 1: start at the y-intercept (0, 1), then go up 2 and right 1 for every step.

Another quick method, best for standard form: find both intercepts, plot them, and connect them. For a full walkthrough with more examples, see graphing linear equations.

Writing a linear equation

From two points

Find the equation of the line through (2, 5) and (4, 9).

Slope: m = (9 − 5) / (4 − 2) = 4 / 2 = 2
Point-slope form with (2, 5): y − 5 = 2(x − 2)
Simplify: y = 2x + 1

From a word problem

A phone plan costs $20 a month plus $5 for each gigabyte of data. Let g be the gigabytes used and C the monthly cost. The fixed part is the intercept and the per-unit part is the slope:

C = 5g + 20

If this month's bill is $45, solve 5g + 20 = 45: subtract 20 to get 5g = 25, then divide by 5. You used 5 GB.

Common mistakes to avoid

Only changing one side. If you subtract 4 on the left, subtract 4 on the right too.
Distributing to only the first term. 2(x − 3) is 2x − 6, not 2x − 3.
Sign errors when moving terms. Moving +3 across the equals sign turns it into −3.
Dividing by a negative carelessly. In −2x = 8, divide by −2 to get x = −4, not 4.
Stopping too early. 3x = 9 isn't solved yet; divide by 3 to get x = 3.
Skipping the check. Substituting your answer back in catches most mistakes in seconds.

Practice problems (with answers)

Try each one before opening the answer.

4x + 9 = 33
Show answer
x = 6 (subtract 9: 4x = 24, divide by 4)
7 − 2x = 15
Show answer
x = −4 (subtract 7: −2x = 8, divide by −2)
6x − 5 = 2x + 11
Show answer
x = 4 (subtract 2x: 4x − 5 = 11, add 5: 4x = 16)
3(2x + 1) = 4x + 13
Show answer
x = 5 (distribute: 6x + 3 = 4x + 13, so 2x = 10)
x/4 − 2 = 1
Show answer
x = 12 (add 2: x/4 = 3, multiply by 4)
Write the equation of the line with slope 4 and y-intercept −3.
Show answer
y = 4x − 3
Rewrite 3x − y = 7 in slope-intercept form.
Show answer
y = 3x − 7 (subtract 3x: −y = −3x + 7, then multiply by −1)
5(x − 2) = 5x − 10
Show answer
Infinitely many solutions (both sides simplify to 5x − 10)

Frequently asked questions

What is a linear equation? Give an example.

A linear equation is an equation in which every variable has an exponent of 1 and variables are never multiplied or divided by each other. Its graph is a straight line. Examples: 3x − 4 = 11, y = 2x + 1, and 2x + 3y = 6.

How do I solve a linear equation?

Simplify each side (distribute and combine like terms), move all variable terms to one side and all constants to the other using inverse operations, then divide by the coefficient of the variable. Whatever you do to one side, do to the other. Finally, substitute your answer back in to check it.

Is 2x + 4y a linear equation?

2x + 4y on its own is a linear expression, not an equation, because it has no equals sign. Once it is set equal to something, such as 2x + 4y = 8, it is a linear equation in two variables: both x and y have an exponent of 1, and its graph is a straight line.

What are the forms of a linear equation?

The three common forms are slope-intercept form (y = mx + b), standard form (Ax + By = C), and point-slope form (y − y₁ = m(x − x₁)). They describe the same line in different ways, and you can convert between them with algebra.

Can a linear equation have no solution or infinitely many solutions?

Yes. If solving leads to a false statement like 6 = 5, the equation has no solution. If it leads to a statement that is always true, like −3 = −3, every value of x works, so there are infinitely many solutions.

What is the difference between a linear equation and a linear function?

A linear function is a rule like f(x) = 2x + 1 that assigns one output to each input; its graph is a non-vertical line. A linear equation is any equation whose graph is a line. Every linear function can be written as a linear equation, but a vertical line such as x = 3 is a linear equation that is not a function.

Where this shows up next

Once solving a single linear equation feels solid, the next steps are graphing lines, solving two equations at once as a system of equations, and working with linear inequalities. Linear equations come back in Algebra 2 and again in calculus, where the tangent line to a curve is a linear equation.

Continue your algebra journey

Mastered linear equations? These topics build directly on them.

Linear Equations Worksheet

Free printable practice: slope, writing equations and converting forms, with answers.

Two-Step Equations Worksheet

Free printable practice: one-step, two-step and multi-step equations with answers.

Graphing Linear Equations

Plot any line from its equation using slope and intercepts.

Systems of Equations

Solve two linear equations at once by substitution, elimination or graphing.

Linear Inequalities

Solve and graph inequalities, and learn when to flip the sign.

Algebra 1 Course

See where linear equations fit in the full Algebra 1 course.

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