12 Systems of Linear Equations and Inequalities

12.1 Solving Systems of Linear Equations by Graphing Objectives 1. Decide whether a given ordered pair is a solution of a system. 2. Solve linear systems by graphing. 3. Solve special systems by graphing. 4. Identify special systems without graphing. 2

Decide Whether a Given Ordered Pair is a Solution A system of linear equations, often called a linear system, consists of two or more linear equations with the same variables. 2x + 3y = 4 3x – y = –5 x + 3y = 1 –y = 4 – 2x x – y = 1 y = 3 or or A solution of a system of linear equations is an ordered pair that makes both equations true at the same time. A solution of an equation is said to satisfy the equation.

Decide Whether a Given Ordered Pair is a Solution Example 1 Is (2,–1) a solution of the system 3x + y = 5 2x – 3y = 7 ? Substitute 2 for x and –1 for y in each equation. 3(2) + (–1) = 5 ? 2(2) – 3(–1) = 7 ? 6 – 1 = 5 ? 4 + 3 = 7 ? 5 = 5 True 7 = 7 True Since (2,–1) satisfies both equations, it is a solution of the system.

Decide Whether a Given Ordered Pair is a Solution Example 1 Is (2,–1) a solution of the system x + 5y = –3 4x + 2y = 1 ? Substitute 2 for x and –1 for y in each equation. 2 + 5(–1) = – 3? 4(2) + 2(–1) = 1? 2 – 5 = –3? 8 – 2 = 1? –3 = –3 True 6 = 1 False (2,–1) is not a solution of this system because it does not satisfy the second equation. 5

Solve Linear Systems by Graphing Example 2 Solve the system of equations by graphing both equations on the same axes. Rewrite each equation in slope-intercept form to graph. –2x + ⅔y = –4 becomes y = 3x – 6 y-intercept (0, – 6); m = 3 5x – y = 8 becomes y = 5x – 8 y-intercept (0, – 8); m = 5

Solve Linear Systems by Graphing Example 2 (concluded) y = 3x – 6 y = 5x – 8 Graph both lines on the same axes and identify where they cross. Because (1,–3) satis-fies both equations, the solution set of this system is {(1,–3)}.

Solve Linear Systems by Graphing CAUTION With the graphing method, it may not be possible to determine from the graph the exact coordinates of the point that represents the solution, particularly if these coordinates are not integers. The graphing method does, however, show geometrically how solutions are found and is useful when approximate answers will suffice.

Solve Special Systems by Graphing Example 3 Solve each system by graphing. 3x + y = 4 6x + 2y = 1 Rewrite each equation in slope-intercept form to graph. 3x + y = 4 becomes y = –3x + 4; y-intercept (0, 4); m = –3 6x + 2y = 1 becomes y = –3x + ½ y-intercept (0, ½); m = –3

Solve Special Systems by Graphing Example 3 (continued) y = –3x + 4 y = –3x + ½ The graphs of these lines are parallel and have no points in common. For such a system, there is no solution.

Solve Special Systems by Graphing Example 3 (continued) Solve each system by graphing. (b) ½x + y = 3 2x + 4y = 12 Rewrite each equation in slope-intercept form to graph. ½x + y = 3 becomes y = –½x + 3; y-intercept (0, 3); m = –½ 2x + 4y = 12 becomes y = –½x + 3; y-intercept (0, 3); m = –½

Solve Special Systems by Graphing Example 3 (concluded) y = – ½x + 3 y = – ½x + 3 The graphs of these two equations are the same line. Thus, every point on the line is a solution of the system, and the solution set contains an infinite number of ordered pairs that satisfy the equations.

Solve Special Systems by Graphing Three Cases for Solutions of Linear Systems with Two Variables 1. The graphs intersect at exactly one point, which gives the (single) ordered-pair solution of the system. The system is consistent, and the equations are independent.

Solve Special Systems by Graphing Three Cases for Solutions of Linear Systems with Two Variables (cont.) 2. The graphs are parallel lines. So, there is no solution and the solution set is . The system is inconsistent and the equations are independent.

Solve Special Systems by Graphing Three Cases for Solutions of Linear Systems with Two Variables (cont.) 3. The graphs are the same line. There is an infinite number of solutions, and the solution set is written in set-builder notation. The system is consistent and the equations are dependent.

Identify Special Systems without Graphing Example 4 Describe the system without graphing. State the number of solutions 3x + 2y = 6 –2y = 3x – 5 Rewrite each equation in slope-intercept form. 16

Identify Special Systems without Graphing Example 4 (continued) Both lines have slope but have different y-intercepts, (0,3) and . Lines with the same slope are parallel, so these equations have graphs that are parallel lines. Thus, the system has no solution. 17