Chapter 3

Systems of Linear Equations

& Inequalities

Solving Systems by

Graphing, Substitution, or

EliminationSections 3.1 & 3.2

A collection of equations in the same variables.

System of Equations

The solution of a system of 2 linear equations in x and y is any ordered pair, (x, y), that satisfies both equations.

The solution (x, y) is also the point of intersection for the graphs of the lines in the system.

The solution of a system…

The ordered pair (2, -1) is the solution of the system below.

y = x – 3 y = 5 – 3x

Example, Pg 156

ACTIVITY 1

Exploring Graphs of

SystemsYOU WILL NEED: graph paper or a graphing calculator

SystemI. Y = 2x – 1 Y = -x + 5

II. Y = 2x – 1 Y = 2x + 1

III. Y =

Y = x + 2

Graph System I at left. ◦ Are there any points

of intersection?◦ Can you find exactly

one solution to the system? If so, what is it?

Repeat for Systems II and III.

Graphing Systems of Equations

4

38 x

4

3

I. Y = 2x – 1 Y = -x + 5

Plug in your equations to Y=

Press Graph

Using the Calculator

Press 2nd, CALC

Select 5: INTERSECT

To find point of Intersection

FIRST CURVE? Press Enter to select the line.

SECOND CURVE? Press Enter to select the 2nd line

GUESS? Move the cursor close to the point of intersection and press Enter

Intersection Point(2, 3)

CLASSIFYING SYSTEMS

OF EQUATIONS

Graphing a system in 2 variables will tell you whether a solution for the system exists.

3 possibilities for a system of 2 linear equations in 2 variables.

Graphing

If a system of equations has at least 1 solution, it is called consistent

◦If a system has exactly one solution, it is called independent(INTERSECTING)

◦If a system has infinitely many solutions, it is called dependent (SAME LINE) (COINCIDING)

If a system does not have a solution, it is called inconsistent. (PARALLEL LINES) (NO SOLUTION)

Graph and Classify each system. Then find the solution from the graph.

x + y = 5 x – 5y = -7

Begin by solving each equation for y.

Example

Graph and find the intersection point like Activity 1.

y = 5 – x y =

Consistent & Independent

USING SUBSTITUTI

ON

2x + y = 33x – 2y = 8

Solve the first equation for y.

EXAMPLE

SUBSTITUTE 3 – 2x into the second equation for y. SOLVE

Substitute 2 for x in either original equation to find y.

Solution: (2, -1)

Check:

Check your Solution!

Try This…Check your Answer

Example

Solving Systems by ELIMINATION

Involves multiplying and combining the equations in a system in order to eliminate a variable.

Elimination Method

Example

Now plug in y = 1 into either of your two original equations.

ASSIGNMENTPg 160-163

Pg 168-170