Solving Systems of Linear Equations by Graphing

Students graph two linear equations, identify their intersection as the solution, and interpret cases with one, no, or infinitely many solutions.

Reviewing Coordinate-Plane Graphing

A coordinate plane has a horizontal x-axis and a vertical y-axis. An ordered pair, written (x, y), tells how far to move from the origin: first horizontally and then vertically. A linear equation graphs as a straight line. For example, y = 2x − 1 is in slope-intercept form, y = mx + b. Its slope is 2, and its y-intercept is −1. Begin at (0, −1). A slope of 2 means rise 2 units and run 1 unit, so another point is (1, 1). Continuing gives (2, 3). Draw a straight line through the points and extend it in both directions. Every point on that line makes the equation true.

Graphing Both Linear Equations

To solve a system by graphing, draw both equations on the same coordinate plane. Use a different color or line style for each graph. Consider the system y = x + 1 and y = −x + 5. For y = x + 1, plot the y-intercept (0, 1), then use the slope 1 to reach points such as (1, 2) and (2, 3). For y = −x + 5, plot (0, 5), then use the slope −1 to reach (1, 4) and (2, 3). Draw both straight lines carefully and extend them across the plane. Because the same coordinate scale is used for both lines, their relationship and shared point can be seen clearly.

Identifying the Intersection Point

The solution of a system is an ordered pair that makes both equations true. On a graph, this solution appears where the two lines intersect. In the system y = x + 1 and y = −x + 5, the lines cross at (2, 3). Therefore, x = 2 and y = 3 solve both equations. Read the x-coordinate first and the y-coordinate second. If an intersection falls between grid lines, estimate its coordinates using the scale and report an approximate solution. For example, a graph might suggest an intersection near (1.5, 2.7). Graphing is useful for seeing the solution, but the result is only as accurate as the drawing, line thickness, grid scale, and coordinate reading.

Checking the Solution Algebraically

A graph can suggest a solution, but substitution confirms it exactly. Check the point (2, 3) in both equations from the system y = x + 1 and y = −x + 5. In the first equation, substitute x = 2 and y = 3: 3 = 2 + 1, which is true. In the second equation, substitute the same values: 3 = −2 + 5, which is also true. Because both statements are true, (2, 3) is the system’s solution. If a point makes only one equation true, it lies on only one of the lines and is not a solution to the system. Algebraic checking is especially important when a graph’s intersection is difficult to read precisely.

Classifying Special Cases

A system of two linear equations can have one solution, no solution, or infinitely many solutions. Lines with different slopes intersect once, so they have one solution. For example, y = x + 1 and y = −x + 5 meet at (2, 3). Distinct lines with the same slope are parallel and never meet, so they have no solution. For example, y = 2x + 1 and y = 2x − 3 are parallel. Equations that produce the same line have every point in common, so they have infinitely many solutions. For example, y = 2x + 1 and 2y = 4x + 2 are equivalent equations. Comparing slopes and y-intercepts helps classify a system before or after graphing it.

Applying Systems to a Context

Systems can model situations involving two unknown quantities. Suppose a school play sells 20 tickets. Student tickets cost $5, adult tickets cost $8, and total sales are $124. Let x represent student tickets and y represent adult tickets. The number equation is x + y = 20, or y = 20 − x. The revenue equation is 5x + 8y = 124, or y = 15.5 − 0.625x. Graph both equations using nonnegative values because negative ticket amounts are impossible. The lines intersect at (12, 8), meaning 12 student tickets and 8 adult tickets were sold. Check the context: 12 + 8 = 20 and 5(12) + 8(8) = 124.