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PPT – Prerequisite Skills Review PowerPoint presentation | free to download - id: 66f71e-MWY3Y

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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 6 X 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)

Lines intersect one solution

Lines are parallel no solution

Lines coincide infinitely 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.

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.

What is the solution of the system graphed below?

- (2, -2)
- (-2, 2)
- No solution
- Infinitely many solutions

Name the Solution

Name the Solution

Name the Solution

How to Use Graphs to Solve Linear Systems

Consider the following system

x y 1 x 2y 5

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 ?

x 2y 5 (1) 2(2) 1 4 5 ?

Since (1 , 2) makes both equations true, then (1

, 2) is the solution to the system of linear

equations.

Solving a system of equations by graphing.

- Let's summarize! There are 3 steps to solving a

system using a graph.

Graph using slope and y intercept or x- and

y-intercepts. Be sure to use a ruler and graph

paper!

Step 1 Graph both equations.

This is the solution! LABEL the solution!

Step 2 Do the graphs intersect?

Substitute the x and y values into both equations

to verify the point is a solution to both

equations.

Step 3 Check your solution.

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 joblets 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. Ill do

slope-intercept form on this one! - y 2x 3
- y 2x 1
- Graph using slope and y-intercept

Graph the equations.

- y 2x 3
- m 2 and b -3
- y 2x 1
- m 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 dont have to graph them!

y 2x 0 y -1x 3

Slope -1/1

Y

Slope 2/1

y-intercept 0

X

Up 2 and right 1

Down 1 and right 1

y-intercept 3

(1,2)

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

y x - 3 y -3x 1

Slope -3/1

Y

Slope 1/1

y-intercept -3

X

y-intercept 1

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

y -2x 4 y 2x 0

Slope 2/1

Y

Slope -2/1

y-intercept 4

X

y-intercept 0

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

Graphing to Solve a Linear System

Using the slope intercept form of these

equations, we can graph them carefully on graph

paper.

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

Lastly, we need to verify our solution is

correct, by substituting (3 , 1).

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 Graphing

Inconsistent 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 its an inconsistent system

Practice Solving by Graphing

Consistent infinite sols

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 -2x3y10

- (-5,0)
- -5-3(0) -5
- -5 -5
- -2(-5)3(0)10
- 1010
- Solution

- (1,4)
- 1-3(4) -5
- 1-12 -5
- -11 -5
- doesnt work in the 1st eqn, no need to check

the 2nd. - Not a solution.

Ex Solve the system graphically. 2x-2y

-8 2x2y4

(-1,3)

Ex Solve the system graphically. 2x4y12 x2y

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-y5 2x-2y9

- 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 8 2y 6x -16

- (3, 1)
- (4, 4)
- No solution
- Infinitely many solutions

You Try It

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.

Problem 1

The two equations in slope-intercept form are

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 .

Problem 2

The two equations in slope-intercept form are

Plot 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.

Problem 3

The two equations in slope-intercept form are

Plot 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

solution (2, 3)

Key Skills

- Solve a system of two linear equations in two

variables graphically.

y 2x 2

y x 1

solution (1, 0)

Key Skills

- Solve a system of two linear equations in two

variables graphically.

y 2x 2

y 2x 4

No solution, why?

Because the 2 lines have the same slope.

Key Skills

TRY THIS

- Solve a system of two linear equations in two

variables graphically.

y 3x 2

solution (-3, -1)

Key Skills

TRY THIS

- Solve a system of two linear equations in two

variables graphically.

2x 3y -12

4x 4y 4

solution (-1.5, -3)

Consider the System

BACK

Graph each system to find the solution

(-3, 1)

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

(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