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.