Graphing Linear Equations

A linear equation describes a relationship with a constant rate of change. Its graph is a straight line made of all ordered pairs that satisfy the equation. In a real situation, identify the independent variable (input) and dependent variable (output), then connect the equation, table, graph, and context.

Slope measures rate of change: m = change in y divided by change in x. For a proportional relationship, the line passes through the origin and has equation y = mx. In slope-intercept form, y = mx + b, m is the slope and b is the initial value or y-intercept.

Example: Graph y = 2x - 3. Plot the y-intercept (0, -3). The slope 2 can be written 2/1, so move up 2 and right 1 to plot another point. Draw a straight line through the points.

You can also make a table. For y = -x + 4, choose x-values, substitute them, and record the ordered pairs. If x = 0, 1, and 4, then y = 4, 3, and 0. Plot (0, 4), (1, 3), and (4, 0).

Linear equations may appear in slope-intercept, point-slope, or standard form. Convert forms when useful. To graph 2x + y = 6, use the intercepts (3, 0) and (0, 6), or rewrite it as y = -2x + 6.

Horizontal lines have slope 0 and equations such as y = 5. Vertical lines have undefined slope and equations such as x = -2. Parallel lines have equal slopes; perpendicular nonvertical lines have negative reciprocal slopes.

A system of two linear equations asks where both equations are true. On a graph, the solution is the intersection point. Intersecting lines have one solution, parallel distinct lines have no solution, and the same line graphed twice has infinitely many solutions.