Prerequisite Skills Review

1.) Simplify: 8r + (-64r)

2.) Solve: 3x + 7(x – 1) = 23

3.) Decide whether the ordered pair (3, -7) is a solution of the equation 5x + y = 8

Section 7.1

Solving Systems by Graphing

What is a system????

Working with 2 equations at one time:

Example:

2x – 3y = 6X + 5y = -12

What is a system of equations?

A system of equations is when you have two or more equations using the same variables.

The solution to the system is the point that satisfies ALL of the equations. This point will be an ordered pair.

When graphing, you will encounter three possibilities.

Intersecting Lines

The point where the lines intersect is your solution.

The solution of this graph is (1, 2)

(1,2)

NUMBER OF SOLUTIONS OF A LINEAR SYSTEM

IDENTIFYING THE NUMBER OF SOLUTIONS

y

x

y

x

Lines intersectone solution

Lines are parallelno solution

y

x

Lines coincideinfinitely many solutions

Parallel Lines

These lines never intersect!

Since the lines never cross, there is NO SOLUTION!

Parallel lines have the same slope with different y-intercepts.

2Slope = = 2

1y-intercept = 2

y-intercept = -1

Coinciding Lines

These lines are the same! Since the lines are on top

of each other, there are INFINITELY MANY SOLUTIONS!

Coinciding lines have the same slope and y-intercepts.

2Slope = = 2

1y-intercept = -1

What is the solution of the system graphed below?

1. (2, -2)

2. (-2, 2)

3. No solution

4. Infinitely many solutions

Name the Solution

Name the Solution

Name the Solution

x – y = –1

x + 2y = 5

How to Use Graphs to Solve Linear Systems

x

yConsider the following system:

(1 , 2)

We must ALWAYS verify that your coordinates actually satisfy both equations.

To do this, we substitute the coordinate (1 , 2) into both equations.

x – y = –1

(1) – (2) = –1 Since (1 , 2) makes both equations true, then (1 , 2) is the solution to the system of linear equations.

x + 2y = 5

(1) + 2(2) =

1 + 4 = 5

Solving a system of equations by graphing.

Let's summarize! There are 3 steps to solving a system using a graph.

Step 1: Graph both equations.

Step 2: Do the graphs intersect?

Step 3: Check your solution.

Graph using slope and y – intercept or x- and y-intercepts. Be sure to use a ruler and graph paper!

This is the solution! LABEL the solution!

Substitute the x and y values into both equations to verify the point is a solution to both equations.

1) Find the solution to the following system:

2x + y = 4

x - y = 2

Graph both equations. I will graph using x- and y-intercepts (plug in zeros).

Graph the ordered pairs.

2x + y = 4(0, 4) and (2, 0)

x – y = 2(0, -2) and (2, 0)

Graph the equations.

2x + y = 4

(0, 4) and (2, 0)

x - y = 2(0, -2) and (2, 0)

Where do the lines intersect?(2, 0)

2x + y = 4

x – y = 2

Check your answer!

To check your answer, plug the point back into both equations.

2x + y = 4 2(2) + (0) = 4

x - y = 2(2) – (0) = 2 Nice job…let’s try another!

2) Find the solution to the following system:

y = 2x – 3

-2x + y = 1

Graph both equations. Put both equations in slope-intercept or standard form. I’ll do slope-intercept form on this one!

y = 2x – 3

y = 2x + 1

Graph using slope and y-intercept

Graph the equations.

y = 2x – 3m = 2 and b = -3

y = 2x + 1m = 2 and b = 1

Where do the lines intersect?No solution!

Notice that the slopes are the same with different y-intercepts. If you recognize this early, you don’t

have to graph them!

y = 2x + 0 & y = -1x + 3

Slope = 2/1y-intercept= 0 Up 2

and right

1

y-intercept= +3

Slope = -1/1

Down 1 and right 1The solution is the point they cross at (1,2)

(1,2)

y = x - 3 & y = -3x + 1

Slope = 1/1y-intercept= -3

y-intercept= +1

Slope = -3/1

The solution is the point they cross at (1,-2)

The solution is the point they cross at (1,2)

y =-2x + 4 & y = 2x + 0

Slope = -2/1y-intercept= 4

y-intercept= 0

Slope = 2/1

Graphing to Solve a Linear System

512 2

23 1

y x

y x

=- +

= -

Solve the following system by graphing: 3x + 6y = 15

–2x + 3y = –3

Using the slope intercept form of these equations, we can graph them carefully on graph paper.

x

y

Start at the y - intercept, then use the slope.

Label the solution!

(3 , 1)

Lastly, we need to verify our solution is correct, by substituting (3 , 1).

Since and , then our solution is correct!( ) ( )3 3 6 1 15+ = ( ) ( )2 3 3 1 3- + =-

Practice – Solving by Graphing

y – x = 1 (0,1) and (-1,0)

y + x = 3 (0,3) and (3,0)

Solution is probably (1,2) …

Check it:

2 – 1 = 1 true

2 + 1 = 3 true

therefore, (1,2) is the solution

(1,2)

Practice – Solving by GraphingInconsistent: no solutions

y = -3x + 5 (0,5) and (3,-4)

y = -3x – 2 (0,-2) and (-2,4)

They look parallel: No solution

Check it:

m1 = m2 = -3

Slopes are equal

therefore it’s an inconsistent system

Practice – Solving by GraphingConsistent: infinite sol’s

3y – 2x = 6 (0,2) and (-3,0)

-12y + 8x = -24 (0,2) and (-3,0)

Looks like a dependant system …

Check it:

divide all terms in the 2nd equation by -4

and it becomes identical to the 1st equation

therefore, consistent, dependant system

(1,2)

Ex: Check whether the ordered pairs are solns. of the system.

x-3y= -5-2x+3y=10

A. (1,4)

1-3(4)= -5

1-12= -5

-11 = -5

*doesn’t work in the 1st eqn, no need to check the 2nd.

Not a solution.

B. (-5,0)

-5-3(0)= -5

-5 = -5

-2(-5)+3(0)=10

10=10

Solution

Ex: Solve the system graphically.2x-2y= -82x+2y=4

(-1,3)

Ex: Solve the system graphically.2x+4y=12x+2y=6

1st eqn:

x-int (6,0)

y-int (0,3) 2ND eqn:

x-int (6,0)

y-int (0,3) What does this mean?

the 2 eqns are for the same line!

¸ many solutions

Ex: Solve graphically: x-y=5 2x-2y=9 1st eqn:

x-int (5,0)

y-int (0,-5) 2nd eqn:

x-int (9/2,0)

y-int (0,-9/2) What do you notice

about the lines? They are parallel! Go

ahead, check the slopes! No solution!

What is the solution of this system?

3x – y = 82y = 6x -16

1. (3, 1)

2. (4, 4)

3. No solution

4. Infinitely many solutions

Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, determine the solution.

1 3 3

3 9 9

.

x y

x y

23

54

5 3

. y x

y x

3 3

2 6

. x y

x y

x

y

The two equations in slope-intercept form are:

y x

y x o r y x

1

31

3

9

9

9

1

31

Plot points for each line.

Draw in the lines.

These two equations represent the same line.

Therefore, this system of equations has infinitely many solutions .

The two equations in slope-intercept form are:

x

y

y x

y x

3

54

3

5Plot points for each line.

Draw in the lines.

This system of equations represents two parallel lines.

This system of equations has no solution because these two lines have no points in common.

x

y

The two equations in slope-intercept form are:y x

y x

3

2 6Plot points for each line.

Draw in the lines.

This system of equations represents two intersecting lines.

The solution to this system of equations is a single point (3,0) .

Key Skills

Solve a system of two linear equations in two variables graphically.

y = 2x 1

y = 12

x + 4 22

4

–2 4 6–4x

y

6

–2

–4

–6

–6

solution: (2, 3)

Key Skills

Solve a system of two linear equations in two variables graphically.

22

4

–2 4 6–4x

y

6

–2

–4

–6

–6

solution:≈ (1, 0)

y + 2x = 2

y + x = 1

Key Skills

Solve a system of two linear equations in two variables graphically.

22

4

–2 4 6–4x

y

6

–2

–4

–6

–6

No solution, why?

y = 2x + 2

y = 2x + 4

Because the 2 lines have the same slope.

Key Skills

Solve a system of two linear equations in two variables graphically.

y = 3x + 2

y = 13

x - 2 22

4

–2 4 6–4x

y

6

–2

–4

–6

–6

solution:≈ (-3, -1)

TRY THIS

Key Skills

Solve a system of two linear equations in two variables graphically.

22

4

–2 4 6–4x

y

6

–2

–4

–6

–6

solution:≈ (-1.5, -3)

2x + 3y = -12

4x – 4y = 4

TRY THIS

Consider the System

5 yx3 yx

x y0 52 35 0

x y0 -32 -13 0

1,4

BACK

Graph each system to find the solution:1.) x + y = -2 2x – 3y = -9

2.) x + y = 4 2x + y = 5

3.) x – y = 5 2x + 3y = 0

4.) y = x + 2 y = -x – 4

5.) x = -2 y = 5

(-3, 1)

(1, 3)

(????)

(????)

(-2, 5)

Check whether the ordered pair is a solution of the system:

1.) 3x + 2y = 4 (2, -1) -x + 3y = -5

2.) 2x + y = 3 (1, 1) or (0, 3) x – 2y = -1

3.) x – y = 3 (-5, -2) or (4, 1) 3x – y = 11