Chapter 3

Section 1

Graphing Systems of Equations

Lesson Preview p. 116Graph each equation.

1. y = 3x – 2 2. y = –x 3. y = – x + 4

Graph each equation. Use one coordinate plane for all three graphs.

4. 2x – y = 1 5. 2x – y = –1 6. x + 2y = 2

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Lesson Preview Answers

1. y = 3x – 2 2. y = –xslope = 3 slope = –1y-intercept = –2 y-intercept = 0

3. y = – x + 4 4. 2x – y = 1

slope = – –y= –2x + 1

y-intercept = 4 y = 2x – 1

5. 2x – y = –1 6. x + 2y = 2–y = –2x – 1 2y = –x + 2y = 2x + 1 y = – x + 1

12 1

2

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Graphing Systems of Equations

Systems of Equations – a set of two or more equations that use the same variable

Linear system – a system where the graph of each equation in a system of two variables is a lineA brace is used to keep the equation system

together

32

3

xy

xy

Solutions to Systems of Linear Equations A solution of a system of equations is a set

of values for the variables that makes all the equations true.

One way to solve linear systems is by graphing -the solution is the point(s) where the graphs intersect

Solve the system by graphing.

x + 3y = 23x + 3y = –6

Check: Show that (–4, 2) makes both equations true.

Graph the equations and find the intersection. The solution appears to be (–4, 2).

x + 3y = 2 3x + 3y = –6(–4) + 3(2) 2 3(–4) + 3(2) –6

(–4) + 6 2 –12 + 6 –6 2 = 2 –6 = –6

3

2

3

1

23

23

xy

xy

yx

2

633

633

xy

xy

yxHere are the equations solved for slope intercept form:

Check Understanding p. 117

Solve by graphing. Check your solution.

2

52

yx

yx

2

52

xy

xygraph

Step1:

Graph the equations (slope-intercept form is nice) Step 2: Locate where the

two graphs intersect

Step 3: Name the coordinate point

(1,3)

Step 4: Check in both equations for accuracy.

55

532

53)1(2

52

yx

22

231

2

yx

Classifying Systems -by the number of solutions (graphing)

Independent – the system has one solution (only one coordinate pair will work for all equations). These lines intersect in one point.

Dependent – the system has no unique solutions (there is more than one coordinate pair that will work). These are graphed as coinciding lines.

Inconsistent – the system has no solution (no coordinate pair will work). These lines are parallel.

Classify the System

Independent

One Solution

Dependent

No Unique Solution

(All points of line are solutions)

Inconsistent

No Solution

Classifying Systems without Graphing-comparing slopes and intercepts

Independent – Different slopes

m ≠

Dependent – Equal slopes Equal y-intercepts

m = b =

Inconsistent – Equal slopes Different y-intercepts

m = b ≠

Step 1: Solve for slope intercept form

Step 2: Write clearly m= and b= for each equation

Step 3: Compare slopes and intercepts

Classify the system without graphing.

Since the slopes are the same, compare the y-intercepts.

y = 3x + 2–6x + 2y = 4

y = 3x + 2 –6x + 2y = 4Rewrite in slope-intercept form. y = 3x + 2

m = 3, b = 2 Find the slope and y-intercept. m = 3, b = 2

Since the y-intercepts are the same, the lines are dependent.

Homework