Foundations: Algebra

Algebra is the largest domain on the SAT Math section, thirteen to fifteen of the forty-four questions, and every one of them is linear: no squares, no roots, no exponentials. That sounds narrow, and it is, which is exactly why the domain rewards fundamentals. The test cannot make a linear equation exotic, so it makes it wordy, or hides it in a graph, or asks about a constant instead of x. Under every disguise sits one of five routines: solve for a variable, read or write a line, solve a system, or handle an inequality.

This lesson is those routines, done slowly and correctly. If you already solve equations comfortably, use it to tighten the two places students lose easy points: keeping the equation balanced when a term moves, and reading slope and intercept from a graph or a sentence without swapping them.

The Medium and Advanced Algebra lessons build directly on these moves, adding context, parameters and the cases where a system has no solution or infinitely many. Nothing there works without what is here.

A linear equation in one variable is any equation that can be arranged as ax + b = c, where a, b and c are numbers and a is not zero. Solving it means finding the one value of x that makes the two sides equal. The tool is inverse operations applied to both sides: subtraction undoes addition, division undoes multiplication, and the order is the reverse of the order of operations. For 4x + 7 = 31, subtract 7 from both sides to get 4x = 24, then divide both sides by 4 to get x = 6. Two complications appear constantly: parentheses, which you clear first with the distributive property so that 3(x - 2) becomes 3x - 6; and variables on both sides, which you collect on one side by adding or subtracting the smaller variable term from both sides. Fractions are cleared by multiplying every term by the common denominator before anything else. Always finish by substituting the answer back in; a check takes ten seconds and catches nearly every arithmetic slip.

A linear equation in two variables describes a line. Slope-intercept form, y = mx + b, is the form the SAT uses most: m is the slope and b is the y-intercept, the y-value where x is zero. The slope measures steepness and direction as the change in y divided by the change in x between any two points on the line, m = (y₂ - y₁)/(x₂ - x₁). A positive slope rises left to right, a negative slope falls, a slope of zero is a horizontal line y = b, and a vertical line x = c has no slope at all. Standard form, Ax + By = C, appears too; convert it by solving for y, which gives slope -A/B and y-intercept C/B. The x-intercept is where y is zero; set y to zero and solve.

To write the equation of a line you need its slope and one point. From two points, compute the slope first, then substitute one point into y = mx + b to find b. If a graph is given, the y-intercept can often be read directly and the slope computed from the intercept and one other grid point. Parallel lines have equal slopes; that fact is tested in Foundations as a direct question and in the Medium lesson as a system with no solution.

In context, the slope is a rate and the intercept is a starting value. If a gym charges a 40 dollar joining fee and 25 dollars a month, the cost after m months is C = 25m + 40: the 25 is attached to the variable because it repeats every month, the 40 stands alone because it happens once. Reading a slope in context means saying what one more unit of x does to y: each additional month adds 25 dollars. This interpretation step is a full third of the Algebra questions on a typical test.

A system of two linear equations asks for the one (x, y) pair that satisfies both. Graphically it is where the two lines cross. Algebraically, substitution works whenever a variable is isolated or easy to isolate: replace it in the other equation, solve for the remaining variable, then substitute back. Elimination, adding or subtracting the equations to cancel a variable, is covered in the Medium lesson, along with what happens when the lines are parallel or identical. At the Foundations level every system has exactly one solution, and the check is to substitute the pair into both equations.

A linear inequality uses <, >, less than or equal to, or greater than or equal to instead of =. Solve it with the same inverse operations, with one rule: multiplying or dividing both sides by a negative number reverses the inequality. Dividing -2x < 8 by -2 gives x > -4. The solution is a range rather than a single number, and the SAT usually asks either for the inequality itself or for which value is or is not in the solution set. A strict inequality excludes the boundary value; a non-strict one includes it.

Distributing to the first term only: 3(x - 2) is 3x - 6, not 3x - 2.

Moving a term without changing its sign. Every term that crosses the equals sign changes sign; saying "subtract 7 from both sides" out loud prevents it.

Swapping slope and intercept in a word problem. The number that repeats per unit is the slope; the one-time amount is the intercept.

Computing slope as run over rise, or subtracting the coordinates in different orders on top and bottom.

Reporting x when the question asked for y, or for x + y, or for 2x. Reread the last sentence of the question before you choose.

Flipping the inequality when adding or subtracting a negative, or forgetting to flip when dividing by one. The rule applies to multiplying or dividing by a negative only.

Skipping the check. Substituting the answer back takes ten seconds and catches the arithmetic slips that account for most lost Algebra points.