SAT Algebra Linear Equations Step-by-Step Guide to Solving f Linear Equations on the Digital SAT (Part 1)

SAT Algebra Linear Equations:  Step-by-Step Guide to Solving Linear Equations on the Digital SAT (Part 1)

Build a Strong Algebra Foundation for a High SAT Math Score

Success on the Digital SAT Math section begins with mastering algebra. Among all the algebra topics tested, linear equations are the most fundamental because they appear directly in equation-solving questions and indirectly in linear functions, graph interpretation, systems of equations, coordinate geometry, mathematical modeling, and many real-world word problems.

Many students believe algebra is about memorizing formulas. In reality, algebra is about logical thinking. Every equation tells a mathematical story, and every solution follows a sequence of logical steps. Once you understand those steps, even difficult-looking SAT questions become manageable.

This guide explains every concept carefully, assuming no prior knowledge beyond basic arithmetic. Each example is solved one step at a time, with an explanation for every operation performed. Instead of simply showing the answer, you'll learn why each step works. By the end of this chapter, you'll be able to solve one-step and two-step linear equations confidently, avoid common mistakes, and build the foundation needed for more advanced SAT algebra topics.

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What Is a Linear Equation?

A linear equation is an equation in which every variable has an exponent of one. When the equation is represented on a graph, it forms a straight line rather than a curve.

Some examples of linear equations are:

• x + 5 = 12

• 3x − 7 = 20

• 4y = 36

• 2a + 9 = 19

Although these equations look different, they all follow exactly the same mathematical principles.

Every linear equation contains an unknown value called a variable. Your goal is to determine the value of the variable that makes the equation true.

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Understanding Variables

A variable is simply a symbol that represents an unknown number.

Instead of writing

□ + 8 = 15

mathematicians write

x + 8 = 15

The letter x can represent any number.

If x equals 7, then

7 + 8 = 15

Both sides are equal, so the equation is true.

The variable does not always have to be x.

It may also be

• y

• a

• b

• m

• n

The letter changes, but the method of solving the equation never changes.

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What Does It Mean to Solve an Equation?

To solve an equation means to find the value of the variable that makes both sides exactly equal.

Imagine an old-fashioned balance scale.

If both sides contain the same weight, the scale remains perfectly balanced.

If you remove weight from one side only, the balance tips.

To keep the balance level, whatever you do to one side must also be done to the other side.

This simple idea is the foundation of all algebra.

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The Golden Rule of Algebra

Whatever operation you perform on one side of an equation must also be performed on the other side.

This rule never changes.

If you add 6 to one side, add 6 to the other side.

If you subtract 10 from one side, subtract 10 from the other side.

If you multiply one side by 4, multiply the other side by 4.

If you divide one side by 7, divide the other side by 7.

Following this rule ensures that both sides remain equal throughout the solution.

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Understanding Inverse Operations

An inverse operation is an operation that reverses another operation.

Operation	Inverse Operation
Addition	Subtraction
Subtraction	Addition
Multiplication	Division
Division	Multiplication

For example,

if 8 has been added,

subtract 8.

If a number has been multiplied by 5,

divide by 5.

Inverse operations allow us to remove numbers one by one until only the variable remains.

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Solving One-Step Linear Equations

One-step equations require only one operation to isolate the variable.

Although these questions are among the easiest on the SAT, learning them thoroughly makes later topics much easier.

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Example 1

Solve

x + 9 = 18

Step 1: Identify the operation attached to the variable.

The variable x has 9 added to it.

Since our goal is to leave x by itself, we must remove the +9.

Step 2: Choose the inverse operation.

The opposite of adding 9 is subtracting 9.

Therefore, subtract 9 from both sides of the equation.

x + 9 − 9 = 18 − 9

Step 3: Simplify both sides.

On the left side,

+9 and −9 cancel each other.

x = 9

On the right side,

18 − 9 = 9

Therefore,

x = 9

Step 4: Check the answer.

Substitute 9 into the original equation.

9 + 9 = 18

18 = 18

Both sides are equal.

The solution is correct.

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Example 2

Solve

x − 14 = 23

Step 1

The variable has 14 subtracted from it.

We must remove the −14.

Step 2

The opposite of subtracting 14 is adding 14.

Add 14 to both sides.

x − 14 + 14 = 23 + 14

Step 3

The −14 and +14 cancel.

x = 37

Step 4

Check.

37 − 14 = 23

23 = 23

The answer is correct.

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Example 3

Solve

6x = 48

Step 1

The variable has been multiplied by 6.

Our goal is to remove the multiplication.

Step 2

The opposite of multiplying by 6 is dividing by 6.

Divide both sides by 6.

6x ÷ 6 = 48 ÷ 6

Step 3

On the left side,

the 6 in the numerator and denominator cancel.

x = 8

On the right side,

48 ÷ 6 = 8

Therefore,

x = 8

Step 4

Check.

6 × 8 = 48

48 = 48

The answer is correct.

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Example 4

Solve

x ÷ 5 = 12

Step 1

The variable has been divided by 5.

We need to undo the division.

Step 2

The opposite of division is multiplication.

Multiply both sides by 5.

(x ÷ 5) × 5 = 12 × 5

Step 3

The 5 in the numerator and denominator cancel.

x = 60

Step 4

Check.

60 ÷ 5 = 12

12 = 12

The solution is correct.

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Solving Two-Step Linear Equations

Two-step equations require removing one operation before removing another.

Always work from the outside toward the variable.

Never try to remove the multiplication before removing the addition or subtraction.

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Example 5

Solve

3x + 7 = 25

Step 1

The variable x has first been multiplied by 3.

After that,

7 has been added.

Since addition happened last,

we remove the addition first.

Step 2

Subtract 7 from both sides.

3x + 7 − 7 = 25 − 7

Step 3

Simplify.

+7 and −7 cancel.

3x = 18

Step 4

The variable is still multiplied by 3.

Undo the multiplication by dividing both sides by 3.

3x ÷ 3 = 18 ÷ 3

Step 5

Simplify.

The 3 cancels.

x = 6

Step 6

Check.

3(6) + 7 = 25

18 + 7 = 25

25 = 25

The solution is correct.

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Example 6

Solve

5x − 20 = 35

Step 1

The variable has 20 subtracted.

Remove the subtraction first.

Add 20 to both sides.

5x − 20 + 20 = 35 + 20

Step 2

Simplify.

−20 and +20 cancel.

5x = 55

Step 3

The variable is multiplied by 5.

Divide both sides by 5.

5x ÷ 5 = 55 ÷ 5

Step 4

Simplify.

x = 11

Step 5

Check.

5 × 11 − 20 = 35

55 − 20 = 35

35 = 35

The answer is correct.

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Example 7

Solve

8x + 16 = 64

Step 1

Subtract 16 from both sides.

8x +16 −16 =64 −16

Step 2

Simplify.

8x =48

Step 3

Divide both sides by 8.

8x ÷8 =48 ÷8

Step 4

Simplify.

x =6

Step 5

Check.

8(6)+16=64

48+16=64

64=64

Correct.

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If an equation says

4x + 12

the multiplication happened first,

then the addition.

When solving,

remove the addition first,

then the multiplication.

Thinking this way makes multi-step equations much easier.

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Working with Negative Numbers

Negative numbers scare many students, but the solving process never changes.

Treat them exactly like positive numbers while paying close attention to the signs.

Example

Solve

−4x = 28

Step 1

The variable is multiplied by −4.

Undo the multiplication by dividing both sides by −4.

−4x ÷ −4 = 28 ÷ −4

Step 2

Simplify.

The −4 cancels.

x = −7

Step 3

Check.

−4(−7)=28

28=28

Correct.

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