3.1 Solving Linear Systems by Graphing

©2001 by R. Villar All Rights Reserved

Solving Linear Systems by Graphing

• • •

System of Equations:

two or more equations using the same variables.

Solving a system means to find the solution to both equations together.

Example: Solve x + y = 5 and 2x – y = 4

Graph each line on one coordinate plane Find the point of intersection.

This solution (the ordered pair) solves both equations.

Example:

Solve the system graphically x + y = 5 2x – y = 4 –x –x –2x –2x

y = –x +5

–y = –2x + 4

y = 2x – 4

Get each equation in y = mx + b form.

the solution is

(3, 2)

What is the solution to the system ?

 

x

2

x

 

y y

  5 4

(3, 2)

How can you check the solution?

Plug it in to both equations of the system.

3

+

2

= 5 √ 2(

3

) –

2

= 4 √

Example:

Solve the system graphically.

x – y = –4 2x + y = –2 –y = –x – 4

y = x + 4

y = –2x – 2

The solution to the system is

(–2, 2)

Example:

Solve the system of linear equations by graphing. x + 2y = 6 2x + 4y = 4 2y = – x + 6 4y = –2x + 4

y = –1/2 x + 3

y = –1/2 x + 1

What is the point of intersection?

Parallel lines do not intersect.

The system has

No Solution

Example:

Solve the system of linear equations by graphing. 4x + 2y = 4 2x + y = 2 2y = – 4x + 4

y = –2x + 2

y = –2x + 2

Notice that these are the same line (one on top of the other).

Where do they intersect?

Everywhere. The system has

infinite solutions

.

Lines are Parallel: If slopes are the same, but y-intercepts are different.

y = 2x + 3 & y = 2x – 2

The system has

no solution.

Lines are the Same: If slopes

and

y-intercepts are the same.

y = 2x + 3 & y = 2x +3

The system has

many solutions.