SAT Algebra — one-variable equations with fractions, decimals and constants

Linear equations in one variable are the first skill of the SAT Algebra domain, and the harder questions on them rarely ask only for x. They mix fractions and decimals into the coefficients, hide a constant such as a, k or c in the equation, and ask which value of that constant leaves the equation with no solution, with infinitely many, or with a given solution. This material trains those structural questions together with the careful arithmetic they depend on.

The first questions solve equations with fractional and decimal coefficients: clearing denominators with the least common multiple, distributing a decimal over a difference, and a service bill with a flat fee and an hourly rate given in dollars and cents. The central part works with constants. After distributing and collecting terms, an equation of the form ax + b = cx + d has exactly one solution when the coefficients of x differ, no solution when they match and the constants do not, and infinitely many when both match. You find the constant that forces each case, including a constant multiplied by a factor outside parentheses and one preceded by a minus sign, decide how many solutions a given equation has, and pick the one equation of four that has no solution.

Two questions look at structure rather than arithmetic: finding a constant from a known solution by substitution, and solving for the expression x + 4 directly instead of for x. One question solves ax + b = cx + d for x in terms of the letters, the general form behind every numerical case.

The quiz has twelve multiple-choice questions with worked explanations, each distractor from a typical slip: a constant not multiplied by the outside factor, a sign lost when dividing by a negative number, the fee forgotten in a bill, x given when an expression was asked for. The flashcards collect the three cases and the working rules. A calculator is allowed throughout the SAT Math section, but these questions are faster by structure.

The material is based on the College Board's published description of the SAT Math section (Algebra domain: linear equations in one variable). It is independent practice and is not produced or endorsed by the College Board.

• Solve linear equations in one variable with fractional and decimal coefficients.
• Find the value of a constant that gives a linear equation no solution or infinitely many solutions.
• Decide how many solutions a linear equation has after simplifying both sides.
• Find a constant from a given solution, and solve ax + b = cx + d for x in terms of the constants.
• Solve for an expression such as x + 4 directly when the question asks for it.

Practice material written by Zestly, based on the College Board's description of the digital SAT Math section, Algebra domain (skill: linear equations in one variable).