3.1 Solving Linear Systems by Graphing

Objective: solve a system of linear equations in two variables by graphing

What are systems of linear equations?

Systems of Linear Equations

• A system of two linear equations in two variables x and y consists of two equations. The coefficients of the terms in the equations can be any real numbers3x – y = 3 Equation 1x + 2y = 8 Equation 2

• A solution of a system of two linear equations in two variables is an ordered pair (x,y) that satisfies both equations.

• When you graph the system, the solution is represented by the point (or points) of intersection of the two lines.

Solve a System by Graphing

y = -x +3 y = 2x + 9

Solve a System by Graphing

3x – y = 3 x + 2y = 8

Solve a System by Graphing

x – 3y = 1 -x + y = -1

Solutions of Systems

• It is also possible for a system to have infinitely many solutions or no solution.

• You can find out how many solutions a linear system has by graphing each equation and analyzing the graphs.

Number of Solutions of a Linear System

• Exactly One Solution: – The graph of the system is a pair of lines that intersect in one point.– the lines have different slopes– the system has exactly one solution

• Infinitely Many Solutions:– The graph of the system is a pair of identical lines– The lines have the same slope and the same y-intercept– The system has infinitely many solutions

• No Solution:– The graph of the system is a pair of parallel lines– The lines have the same slope and different y-intercepts– The system has no solution

Number of Solutions of a Linear System

Exactly One Solution Infinitely Many Solutions No Solution

Tell how many solutions the linear system has.

2x – y = 1-4x + 2y = -2

Tell how many solutions the linear system has.

x + 2y = 4x + 2y = 1

Tell how many solutions the linear system has.

x – 5y = 5x + 5y = 5

Basketball

Christie played in a basketball game in which she scored a total of 21 points. In the game, she made twice as many two-point shots as three-point shots. How many of each type of shot did Christie make?