Linear equation
Also known as: first-degree equation, first-degree equations, linear equations
A linear equation is an equation whose variables appear only to the first power, with no products of variables, roots, or variables in denominators. Graphed in two variables, a linear equation always produces a straight line.
A linear equation is one in which every variable has an exponent of 1. In one variable it looks like 3x + 7 = 22, and solving it means isolating x — subtract 7, divide by 3, and x = 5. In two variables, such as 2x + 3y = 12, the solutions form an infinite set of ordered pairs that plot as a straight line. Equations containing x², √x, xy, or 1/x are not linear.
Three forms of a two-variable linear equation come up constantly. Slope-intercept form, y = mx + b, gives the slope m and the y-intercept b directly, and is the fastest form for graphing. Standard form, Ax + By = C, makes the intercepts easy to find: set y = 0 to get the x-intercept and x = 0 to get the y-intercept. Point-slope form, y − y₁ = m(x − x₁), is the natural choice when you know one point and the slope. Slope itself is the rise over the run, m = (y₂ − y₁) / (x₂ − x₁), and it stays constant everywhere along the line — that constant rate of change is what makes the graph straight.
Two lines are parallel when their slopes are equal and perpendicular when their slopes are negative reciprocals, so a line perpendicular to y = 2x + 1 has slope −1/2. A horizontal line has slope 0 and the form y = c; a vertical line has undefined slope and the form x = c. Solving a system of two linear equations means finding the point where the lines cross, using substitution, elimination, or a graph.
Linear equations are among the most heavily tested topics on the SAT, ACT, and CLT. Expect to convert between forms, find slope from two points or a graph, write the equation of a line through given conditions, solve systems, and translate a word problem into a linear model where the slope is a rate and the intercept a starting value.
Key takeaways
- A linear equation has variables only to the first power and graphs as a straight line in two variables.
- Slope-intercept form is y = mx + b; standard form is Ax + By = C; point-slope form is y − y₁ = m(x − x₁).
- Slope is the constant rate of change, m = (y₂ − y₁) / (x₂ − x₁).
- Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.
- The SAT, ACT, and CLT all test converting between forms, solving systems, and building linear models from word problems.