Linear equations in one variable

Solve, interpret and build linear equations with one variable, including ones with no solution or infinitely many solutions. It is part of Algebra in the Math section. The book teaches it in Chapter 16, Linear Equations in One Variable, with 16 practice questions, and tests it 6 more times across the two full-length practice tests.

About Chapter 16

A linear equation in one variable has a single unknown, usually x, and no power of it higher than 1: no x^2, no √x and no x in a denominator. An equation such as 5(x − 2) = 3x + 4 is the most basic algebra on the test. It appears as a stand-alone question, inside word problems, and as the last step of systems, geometry and data questions. This chapter builds a fast, reliable method, then covers the variations that make these equations hard: equations with no solution or infinitely many solutions, unknown constants, and questions that ask for an expression instead of x.

What the question looks like

Most questions in this skill take one of five forms:

  • Solve. An equation is given, often with parentheses, fractions or decimals, and you find x, either from choices or as a student-produced response.
  • Solve for an expression. For example, "If 3x + 2 = 11, what is the value of 6x + 4?"
  • Count solutions. "How many solutions does the given equation have?" The choices are usually zero, exactly one, exactly two and infinitely many.
  • Find a constant. "In the given equation, k is a constant. The equation has no solution. What is the value of k?"
  • Model a context. A short situation with a fixed amount and a rate. You write the equation, solve it or both.

How to recognize it

The equation has one variable, and every term that contains it is a number times the variable, or will be once you distribute and clear fractions. If you see x^2, a variable in an exponent or a variable under a radical, you are in Chapter 21 instead. If you see two variables, such as x and y, you are in Chapter 17 or 18.

Letters such as a, b and k are constants (also called parameters) that stand for fixed numbers. They make a question harder without changing the skill.

In a word problem, look for a starting amount plus or minus a rate times an unknown, such as a fee plus a monthly charge or a balance minus spending per week.

Common traps

  • Distributing to only the first term: A number in front of parentheses touched only one term inside. Recheck each product.
  • Dropping a negative sign: A minus sign stands in front of parentheses, as in −2(x − 5). The second term becomes +10.
  • Clearing only the fractions: A constant without a denominator was not multiplied by the LCD.
  • Moving a term with the wrong sign: A term changed sides but kept its sign. Add or subtract the same amount on both sides.
  • Answering x when the question asks for an expression: The last line of the question names something other than x.
  • Choosing "zero" whenever x cancels: The leftover statement is true, which means infinitely many solutions.
  • Mixing up the conditions for none and infinitely many: You matched the constants when the question said "no solution."
  • Reporting a shared total instead of the unknown: In a context, the answer is the quantity you named as the variable, in its units.

Video lessons

Two short lessons from independent YouTube creators, chosen for this skill. The notes are ours; the videos belong to their creators, who are not affiliated with this site.

  • Linear equations in one variable (Crunch Math, 7:27). Watch for: Clearing parentheses and fractions, and spotting no-solution and infinite-solution cases.
  • Linear word problems (Penguin SAT Prep , 26:00). Watch for: Turning a word problem into an equation before solving.

Practice this skill

The full method, worked examples and 16 practice questions with every answer choice explained are in Chapter 16 of the book and in the study workspace, where an AI coach can talk you through each one.

Independence

Skill by Skill Prep is independent. SAT® is a trademark registered by the College Board, which is not affiliated with, and does not endorse, this website or book. Bluebook™ is a trademark of the College Board. Desmos is a trademark of Desmos Studio PBC. Practice results and ranges are study estimates, not official scores or admission predictions.

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