Solving Systems by Graphing - Algebra I

Solving Systems by Graphing

In this lesson, we will explore how to solve systems of equations by graphing. Graphing is a visual method that involves plotting each equation on the same coordinate plane and identifying the point(s) where the lines intersect, which represents the solution(s) to the system.

Outline

Definition of Systems of Equations
Graphing Method for Solving Systems
Examples of Solving Systems by Graphing
Exercises with Solutions
Homework
Revision

Definition of Systems of Equations

A system of equations is a set of two or more equations with the same variables. The solution to the system is the set of values that satisfy all the equations simultaneously. Solving by graphing involves plotting each equation on the coordinate plane and finding the intersection points.

Graphing Method for Solving Systems

To solve a system of equations by graphing:

Convert each equation to slope-intercept form, y = mx + b, if necessary.
Plot the lines represented by each equation on the same coordinate plane.
Identify the point(s) where the lines intersect. This point is the solution to the system.

Examples

Example 1

Solve the system:

1. y = 2x + 1

2. y = -x + 3

Solution: Graph both lines. The point of intersection is (1, 3).

Example 2

Solve the system:

1. y = -x + 4

2. y = x - 2

Solution: Graph both lines. The point of intersection is (3, 1).

Example 3

Solve the system:

1. y = 0.5x - 1

2. y = -0.5x + 2

Solution: Graph both lines. The point of intersection is (2, 0).

Example 4

Solve the system:

1. 2y = 3x - 6

2. y = x + 1

Solution: Convert the first equation to y = 1.5x - 3 and graph both lines. The point of intersection is (3, 4.5).

Example 5

Solve the system:

1. y = 3x + 2

2. y = -2x + 5

Solution: Graph both lines. The point of intersection is (1, 5).

Example 6

Solve the system:

1. y = -0.25x + 2.5

2. y = 0.75x - 0.5

Solution: Graph both lines. The point of intersection is (1, 2.25).