Math · Algebra

Linear equations — Digital SAT Math practice

Solve and set up equations with one variable.

Key terms

Linear equation: An algebraic statement where the highest power of the variable is one.
Inverse operation: A math move that undoes another, like subtracting to cancel addition.
Like terms: Parts of an expression with identical variables raised to the same powers.
Parameter: A letter representing a fixed number in an equation, treated as a constant.
Identity equation: A linear equation true for all real numbers, yielding infinite solutions.
Contradiction: A false statement like 3 = 5 that results when solving with no solution.

Forgetting to distribute to every term inside parentheses leads to incorrect coefficients.
Moving a term across the equals sign without changing its sign.
Ignoring negative signs when distributing causes sign errors in the final answer.
Assuming all linear equations have exactly one solution misses cases with zero or infinite answers.

A student solves 3x − 5 = 16 and writes 3x = 11. What went wrong?

They subtracted 5 instead of adding it. To undo subtracting 5, add 5 to both sides: 3x = 21, so x = 7.

Watch out: Using the same operation that is already in the equation instead of its inverse, or applying it to only one side.

What is the correct sequence of steps to solve 5(x + 2) - 3x = 4x + 10?

Distribute first, then combine like terms on each side, and finally isolate the variable using inverse operations.

Watch out: Combining terms before distributing or skipping the distribution step leads to incorrect coefficients for the variable.

Without solving, determine how many solutions exist for 3x + 6 = 3(x + 2). Explain your reasoning based on the final simplified statement.

Infinitely many solutions. Simplifying gives 3x + 6 = 3x + 6, and subtracting 3x leaves 6 = 6, which is always true.

Watch out: Stopping at an intermediate step like 3x + 6 = 3x + 6 without simplifying further misses that the variable cancels out completely.

Translate into an equation: “Four less than three times a number is twenty.”

3x − 4 = 20. Three times a number is 3x, and less than reverses the order, so four less than 3x is 3x − 4.

Watch out: Writing 4 − 3x. In “A less than B,” the B comes first: B − A.

Solve for x: 3(x - 2) + 4 = 2x + 5.

Show answer and explanation

Answer: 7

3x - 6 + 4 = 2x + 5, so 3x - 2 = 2x + 5 and x = 7.

A number increased by 25% is equal to 45 less than twice the number. What is the number?

Show answer and explanation

Answer: 60

1.25n = 2n - 45, so 0.75n = 45 and n = 60.