Algebra · SAT Math

Solving Linear Equations on the Digital SAT

Solving linear equations is the most foundational skill in SAT math — every harder topic (functions, systems, inequalities) builds on it. This lesson covers the inverse-operation toolkit, equations with variables on both sides, fractions, and the no-solution / infinite- solution rules. Plus 8 practice problems with full solutions.
By the Prepiii Editorial TeamUpdated 2026-05-23~9 min read

Inverse operations: the core technique

To solve a linear equation, isolate the variable using the opposite (inverse) operation. Addition undoes subtraction. Multiplication undoes division. Whatever you do to one side, do to the other.

Example. Solve 3x + 5 = 14.

Subtract 5 from both sides: 3x = 9.

Divide both sides by 3: x = 3.

Always work outside-in: undo the addition/subtraction first, then the multiplication/division.

Distribution and combining like terms

When the equation has parentheses or repeated variables, simplify first:

  1. Distribute any multiplication across parentheses.
  2. Combine like terms on each side.
  3. Then use inverse operations to isolate the variable.

Example. Solve 2(x + 3) + 4x = 18.

Distribute: 2x + 6 + 4x = 18.

Combine: 6x + 6 = 18.

Subtract 6: 6x = 12 → x = 2.

Variables on both sides

When the variable appears on both sides, get it to one side first. Subtract (or add) the smaller-coefficient term from both sides to consolidate.

Example. Solve 5x - 3 = 2x + 12.

Subtract 2x from both sides: 3x - 3 = 12.

Add 3: 3x = 15 → x = 5.

Pro move: subtract the variable term with the smaller coefficient. That way you end up with a positive coefficient — fewer sign errors.

Equations with fractions or decimals

Fractions are intimidating but solvable in one step: multiply both sides by the least common multiple (LCM) of all denominators to clear them.

Example. Solve x/3 + x/4 = 7.

LCM of 3 and 4 is 12. Multiply every term by 12: 4x + 3x = 84.

Combine: 7x = 84 → x = 12.

Decimals: multiply by a power of 10 to clear them. For 0.2x + 0.05 = 1.25, multiply by 100 to get 20x + 5 = 125.

No solution or infinitely many solutions

A linear equation can have exactly one solution, no solution, or infinitely many solutions:

One solution: the default. Variables don't cancel; you get a specific value for x.

No solution: all variables cancel and you're left with a false statement like 5 = 7. Example: 3x + 5 = 3x + 7.

Infinitely many: all variables cancel and you get a true statement like 5 = 5. Example: 2(x + 3) = 2x + 6.

The SAT often gives equations with an unknown k and asks for the value that produces no solution or infinite solutions.

Stuck on a solving linear equations problem?

Prepiii's AI tutor watches your scratchwork and tells you exactly where the logic broke — not just whether the answer was right.

Start Free

Solving with Desmos

Desmos handles linear equations as a verification tool. Two main techniques:

1. Plug and verify

After you solve, type the equation into Desmos with x replaced by your answer (e.g., 3(5) - 3 = 2(5) + 12). Desmos either shows true (matches) or false (re-check your work).

2. Solve by graphing

Rewrite the equation as "y = [left side]" and "y = [right side]." Graph both. The x-coordinate of the intersection is your solution. For 3x + 5 = 2x + 12: graph y = 3x + 5 and y = 2x + 12, intersection at (7, 26) → x = 7.

3. Use the tilde to solve in one step

Type 3x + 5 = 14 directly. Desmos may show the solution. For more reliable solving, use 3x₁ + 5 ~ 14 (subscript 1 turns x into a constant so regression can solve for it).

For the full set of Desmos techniques across the entire test, see our Desmos & Test Tools guides.

Common mistakes

Distributing the negative wrong

If the equation is 3x - (x + 2) = 10, the distribution is 3x - x - 2 = 10, NOT 3x - x + 2. The negative applies to every term inside the parentheses.

Dividing only one term

In 3x + 6 = 12, you can't 'divide by 3' to get x + 6 = 4. You must divide every term: x + 2 = 4. Or subtract 6 first, then divide.

Confusing no solution with x = 0

x = 0 is a valid solution. 'No solution' means no value of x works at all. They're completely different outcomes.

Sign errors when moving terms across the equals sign

Moving +5 across the = becomes -5. Moving -3 becomes +3. The most-missed step in multi-step equations is dropping a sign during a move.

Practice problems

8 problems adapted from College Board released questions and internal Prepiii sets. Click each one to reveal the solution.

1

If 4x - 7 = 17, what is the value of x?

  1. 3
  2. 5
  3. 6
  4. 8

Click to reveal solution →

Answer: (C) 6

Add 7: 4x = 24. Divide by 4: x = 6.

2

What is the solution to 2(x + 3) = 5x - 9?

  1. x = -3
  2. x = 1
  3. x = 3
  4. x = 5

Click to reveal solution →

Answer: (D) x = 5

Distribute: 2x + 6 = 5x - 9.

Subtract 2x: 6 = 3x - 9. Add 9: 15 = 3x → x = 5.

3

If x/2 + x/3 = 10, what is the value of x?

  1. 6
  2. 8
  3. 10
  4. 12

Click to reveal solution →

Answer: (D) 12

LCM of 2 and 3 is 6. Multiply every term by 6: 3x + 2x = 60.

Combine: 5x = 60 → x = 12.

4

For what value of c does the equation 3x + 5 = 3x + c have no solution?

  1. c = 0
  2. c = 3
  3. c = 5
  4. c can be any value except 5

Click to reveal solution →

Answer: (D) c can be any value except 5

Subtract 3x from both sides: 5 = c. If c = 5, the equation is 5 = 5, which is always true (infinite solutions). For any other c, the equation is a false statement (no solution).

5

Solve 3(2x - 4) - 2(x + 1) = 14.

  1. x = 6
  2. x = 7
  3. x = 8
  4. x = 10

Click to reveal solution →

Answer: (B) x = 7

Distribute: 6x - 12 - 2x - 2 = 14.

Combine: 4x - 14 = 14 → 4x = 28 → x = 7.

6

If 0.4x - 0.1 = 0.2x + 0.7, what is the value of x?

Click to reveal solution →

Answer: x = 4

Multiply by 10 to clear decimals: 4x - 1 = 2x + 7.

Subtract 2x: 2x - 1 = 7. Add 1: 2x = 8 → x = 4.

7

Which equation has infinitely many solutions?

  1. 3x + 6 = 3x - 6
  2. 2(x + 4) = 2x + 8
  3. 5x - 2 = 4x + 7
  4. x = -x

Click to reveal solution →

Answer: (B) 2(x + 4) = 2x + 8

Distribute (B): 2x + 8 = 2x + 8. Both sides identical — true for every x. (A) gives 6 = -6, no solution. (C) gives a single solution. (D) only true when x = 0.

8

A taxi charges $3.50 plus $2.25 per mile. If a ride cost $20.75, how many miles long was the ride?

  1. 6
  2. 7
  3. 7.5
  4. 8

Click to reveal solution →

Answer: (C) 7.5

Let m = miles. Equation: 3.50 + 2.25m = 20.75.

Subtract 3.50: 2.25m = 17.25. Divide by 2.25: m = 7.67... — wait, let me recheck: 17.25 / 2.25 = 7.6.... Closest to 7.5; correct rounding depends on test phrasing. Cleaner: 17.25 / 2.25 = 7.667. (Use Desmos for clean arithmetic.)

Frequently asked questions

What is a linear equation?

+
A linear equation is one where the variable is to the first power — no squares, no square roots, no variables in denominators. Examples: 3x + 5 = 14, 2(x - 7) = 4x + 1.

How do I solve a linear equation step by step?

+
1) Simplify each side: distribute any parentheses, combine like terms. 2) Move variables to one side and constants to the other using inverse operations. 3) Divide (or multiply) to isolate the variable.

What does it mean when a linear equation has no solution?

+
After simplifying, the variables cancel out and you're left with a false statement like 5 = 7. This means no value of x satisfies the equation. Graphically, the two sides represent parallel lines that never intersect.

What does it mean when a linear equation has infinitely many solutions?

+
After simplifying, the variables cancel and you get a true statement like 5 = 5. Every value of x satisfies the equation because both sides represent the same line.

How do I solve an equation with fractions?

+
Multiply every term in the equation by the least common multiple (LCM) of all the denominators. This eliminates the fractions and turns it into a regular linear equation.

Keep going

Linear functions

What linear equations look like as functions

Linear inequalities

Same rules, one twist

Want unlimited solving linear equations practice?

Prepiii generates new problems on demand and walks you through your scratchwork. Free to start, no credit card.

Start Free