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SAT Math: Algebra

Algebra is the single largest content domain on the Digital SAT Math section, making up roughly 35% of all questions. It covers linear equations and inequalities in one and two variables, systems of linear equations, and the translation of real-world scenarios into algebraic expressions.

Because these question types recur with predictable structure, Algebra is often the domain where focused practice produces the fastest, most reliable improvement.

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Linear equations in one variable

These questions ask you to solve for an unknown in an equation involving only addition, subtraction, multiplication, division, and distribution — no exponents on the variable. The key skill is careful, methodical isolation of the variable: distribute first, combine like terms on each side, then move variable terms to one side and constants to the other.

Watch for equations with variables on both sides or with fractions; clearing denominators by multiplying both sides by the least common denominator early often prevents arithmetic errors later in the problem.

Worked example

Solve for x: 3(x − 4) + 5 = 2x + 7

  1. Distribute: 3x − 12 + 5 = 2x + 7
  2. Combine like terms: 3x − 7 = 2x + 7
  3. Subtract 2x from both sides: x − 7 = 7
  4. Add 7 to both sides: x = 14

Answer: x = 14

Linear equations and functions in two variables

Two-variable linear relationships appear as equations (y = mx + b), as tables of x-y pairs, and as graphs. You should be comfortable converting between all three forms quickly. The slope m represents a constant rate of change, and the y-intercept b represents the value of y when x = 0.

In word problems, slope is often a rate (dollars per hour, feet per second) and the y-intercept is often a starting value (an initial fee, a starting height). Recognizing which number in a problem plays which role is usually the fastest path to setting up the equation correctly.

Systems of linear equations

A system of two linear equations can be solved by substitution, elimination, or by recognizing special cases. If the two lines have the same slope but different y-intercepts, the system has no solution (the lines are parallel). If the two equations represent the exact same line, the system has infinitely many solutions.

The SAT frequently tests this conceptual case directly: given a system with a variable coefficient, it asks for the value that makes the system have no solution or infinitely many solutions, which requires setting the slopes (and, for infinite solutions, the intercepts) equal rather than solving for a single intersection point.

Worked example

For what value of k does the system 4x + 6y = 12 and kx + 9y = 18 have infinitely many solutions?

  1. Infinitely many solutions requires the second equation to be a multiple of the first.
  2. Compare the y-coefficients: 9 is 1.5 times 6, so the scale factor is 1.5.
  3. Apply that factor to the x-coefficient: k = 4 × 1.5 = 6.
  4. Check the constants: 12 × 1.5 = 18, which matches, confirming the equations represent the same line.

Answer: k = 6

Linear inequalities

Inequalities are solved the same way as equations, with one crucial rule: multiplying or dividing both sides by a negative number flips the inequality sign. Many SAT inequality questions involve a real-world constraint, such as a maximum budget or a minimum number of items, and ask you to identify which inequality correctly models the scenario before solving it.

Systems of inequalities (two inequalities graphed together) ask you to identify a point or region that satisfies both constraints simultaneously — check each candidate point against both inequalities rather than trying to visualize the shaded region if a graph isn't provided.

Translating word problems into algebra

Many Algebra questions are word problems that require you to translate a description into an equation before doing any solving. Underline the specific numbers and their units, identify what the question is actually asking for, and assign a variable to that unknown quantity before writing anything else.

Common phrases have consistent translations: 'more than' and 'increased by' mean addition, 'less than' means subtraction with the order reversed (five less than x is x − 5, not 5 − x), 'times' or 'of' means multiplication, and 'per' signals a rate that multiplies a variable.

Common mistakes to avoid

  • Reversing subtraction order in phrases like 'five less than a number,' writing 5 − x instead of the correct x − 5.
  • Forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number.
  • Distributing incorrectly by forgetting to multiply every term inside parentheses, especially with a negative sign out front.
  • Confusing slope and y-intercept in context, such as treating a starting fee as the rate of change.
  • Assuming a system with matching slopes always has no solution, without checking whether the y-intercepts also match (which would mean infinitely many solutions instead).
  • Solving for the wrong variable because the problem's question is misread after a long word-problem setup.

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