Table of Contents Solve Systems by Graphing Solve Systems by Substitution Solve Systems by...

Table of Contents

Solve Systems by Graphing

Solve Systems by Substitution

Solve Systems by Elimination

Choosing your Strategy

Writing Systems to Model Situations

Solving Systems of Inequalities

Teacher note:Many of the Responder Questions have boxes over the answers. Have the students take some time to solve the system prior to showing them the multiple choice answers by clicking on the box, so they do not just substitute in the answers.T

Click on topic to go to that section.

Strategy One:Graphing

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Some vocabulary...

The "solution" to a system is an ordered pair that will work in each equation. One way to find the solution is to graph the equations on the same coordinate plane and find the point of intersection.

A "system" is two or more linear equations.

Suppose you are walking to school. Your friend is 5 blocks ahead of you. You can walk two blocks per minute, your friend can walk one block per minute. How many minutes will it take for you to catch up with your friend?

Consider this...

Time (min.)

Friend's distance from your start (blocks)

Your distance from your start(blocks)

5 10 10

First, make a table to represent the problem.

Next, plot the points on a graph.

Time (min.)

Friend's distance from your start (blocks)

Your distance from your start(blocks)

5 10 10

The point where they intersect is the solution to the system.

Time (min.)

(5,10) is the solution. In the context of the

problem this means after 5 minutes, you will

meet your friend at block 10.

Solve the system of equations graphically.

y = 2x -3y = x - 1

2x + y = 3x - 2y = 4

3x + y = 11x - 2y = 6

Solve using graphing

y = 4x+6movey = -3x-1move

Write the equation forthe green dashed line

Write the equation forthe blue solid line

What is this pointof intersection?

(move the hand!)(-1, 2)

( , )-1 2

y = 4x+6y = -3x-1

Now take the ordered pair we just found and substitute it into the equation to prove that it is a solution for both lines.

y = 2x + 3

Solve by Graphing

y = -4x - 3

(-1,1)

y= x - 4y= -3x + 4

Solve by Graphing

(2,-2)

What's the problem here?

y= 2x - 4y= 2x + 4

Parallel lines do not intersect!

Therefore there is no solution.

No ordered pair that will work in BOTH equations

( )click to reveal

click to reveal

2y = -4x + 102 2 y = -2x + 5

2x + y = 5 -2x -2x y = -2x + 5

Solve by Graphing

First - transform the equations into y = mx + b form (slope-intercept form)

Now graph the two transformed lines.

2y = 10 -4x becomesy = -2x + 5

2x + y = 5 becomesy = -2x + 5

What's the problem?

The equations transform to the same line.

So we have infinitely

many solutions.

click to reveal click to reveal

1 Solve the system by graphing.y = -x + 4y = 2x +1

A (3,1)

B (1,3)

C (-1,3)

D no solution

Click for multiple choice answers.

2 Solve the system by graphing.y = 0.5x - 1y = -0.5x -1

A (0,-1)

B (0,0)

C infinitely many

D no solution

3 Solve the system by graphing.2x + y = 3x - 2y = 4

A (2,4)

B (0.4, 2.2)

C (2, -1)

D no solution

4 Solve the system by graphing.y = 3x + 3y = 3x - 3

A (0,0)

B (3,3)

C infinitely many

D no solution

5 Solve the system by graphing.y = 3x + 44y = 12x + 16

A (3,4)

B (-3,-4)

C infinitely many

D no solution

6 On the accompanying set of axes, graph and label the following lines:  y=5 x = - 4 y = x+5

Calculate the area, in square units, of the triangle formed by the three points of intersection.

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Strategy Two:Substitution

y = x + 6.1y = -2x - 1.4

Graphing can be inefficient or approximate.

Another way to solve a system is to use substitution.

Substitution allows you to create a one variable equation.

Substitution Explanation

Solve the system using substitution. Why was it difficult to solve this system by graphing?

y = x + 6.1y = -2x - 1.4

y = -2x - 1.4 -start with one equationx + 6.1 = -2x - 1.4 -substitute x + 6.1 for y in equation+2x -6.1 +2x - 6.1 3x = -7.5 -solve for x  x = -2.5

Substitute -2.5 for x in either equation and solve for y. y = x + 6.1 y = (-2.5) + 6.1 y = 3.6

Since x = -2.5 and y = 3.6, the solution is (-2.5, 3.6)

CHECK: See if (-2.5, 3.6) satisfies the other equation. y = -2x - 1.43.6 = -2(-2.5) - 1.43.6 = 5 - 1.43.6 = 3.6

+ 3x = 21-3 y

y = -2x +14

Solve the system using substitution.

= -y - 3x

x = -5y - 39

Solve the system using substitution.

Examine each system of equations.Which variable would you choose to substitute?Why?

y = 4x - 9.6y = -2x + 9

y = -3x7x - y = 42

y = 4x + 1x = 4y + 1

7 Examine the system of equations. Which variable would you substitute?

2x + y = 52y = 10 - 4x

2y - 8 = xy + 2x = 4

x - y = 202x + 3y = 0

Sometimes you need to rewrite one of the equations so that you can use the substitution method. For example:

The system: Is equivalent to:3x -y = 5 y = 3x -52x + 5y = -8 2x + 5y = -8

Using substitution you now have: 2x + 5(3x-5) = -8  -solve for x2x + 15x - 25 = -8 -distribute the 5  17x - 25 = -8 -combine x's  17x = 17 -at 25 to both sides  x = 1 - divide by 17

Substitute x = 1 into one of the equations.2(1) + 5y = -8 2 + 5y = -8 5y = -10 y = -2

The ordered pair (1,-2) satisfies both equations in the original system. 3x -y = 5    2x + 5y = -83(1) - (-2) = 5 2(1) + 5(-2) = -8 3 + 2 = 5  2 - 10 = -8  -8 = -8

Your class of 22 is going on a trip. There are four drivers and two types of vehicles, vans and cars. The vans seat six people, and the cars seat four people, including drivers. How many vans and cars does the class need for the trip?

Let v = the number of vansand c = the number of cars

Set up the system:

 Drivers: v + c = 4 People: 6v + 4c = 22

Solve the system by substitution.  v + c = 4  -solve the first equation for v.    v = -c + 4  -substitute -c + 4 for v in the 6(-c + 4) + 4c = 22  second equation -6c + 24 + 4c = 22  -solve for c  -2c + 24 = 22  -2c = -2  c = 1

  v + c = 4   v + 1 = 4 -substitute for c in the 1st equation  v = 3 -solve for v

Since c = 1 and v = 3, they should use 1 car and 3 vans.

Check the solution in the equations:  v + c = 4 6v + 4c = 22 3 + 1 = 4  6(3) + 4(1) = 22  4 = 4  18 + 4 = 22  22 = 22

Now solve this system using substitution. What happens? x + y = 6 5x + 5y = 10

x + y = 6 -solve the first equation for x  x = 6 - y5(6 - y) + 5y = 10 -substitute 6 - y for x in 2nd equation 30 - 5y + 5y = 10 -solve for y 30 = 10 -FALSE!

Since 30 = 10 is a false statement, the system has no solution.

Now solve this system using substitution. What happens? x + 4y = -3 2x + 8y = -6

x + 4y = -3 - solve the first equation for x  x = -3 - 4y2(-3 - 4y) + 8y = -6 - sub. -3 - 4y for x in 2nd equation -6 - 8y + 8y = -6 - solve for y -6 = -6   - TRUE! - there are infinitely many solutions

How can you quickly decide the number of solutions a system has?

1 Solution Different slopes

No Solution Same slope; different y-intercept (Parallel Lines)

Infinitely Many Same slope; same y-intercept (Same Line)

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

A 1 solution

B no solution

C infinitely many solutions

11  3x + 3y = 8 y = x

A 1 solution

B no solution

12  y = 4x  2x - 0.5y = 0

A 1 solution

B no solution

13 3x + y = 5 6x + 2y = 1

A 1 solution

B no solution

14  y = 2x - 7 y = 3x + 8

A 1 solution

B no solution

15 Solve each system by substitution.y = x - 3y = -x + 5

A (4,9)

B (-4,-9)

C (4,1)

D (1,4)

16 Solve each system by substitution.y = x - 6y = -4

A (-10,-4)

B (-4,2)

C (2,-4)

D (10,4)

17 Solve each system by substitution.y + 2x = -14y = 2x + 18

A (1,20)

B (1,18)

C (8,-2)

D (-8,2)

18 Solve each system by substitution.4x = -5y + 50x = 2y - 7

A (6,6.5)

B (5,6)

C (4,5)

D (6,5)

19 Solve each system by substitution.y = -3x + 23-y + 4x = 19

A (6,5)

B (-7,5)

C (42,-103)

D (6,-5)

Strategy Three:Elimination

When both linear equations of a system are in Standard Form, Ax + By = C, you can solve the system using elimination.

You can add or subtract the equations to eliminate a variable.

How do you decide which variable to eliminate?

First, look to see if one variable has the same or opposite coefficients. If so, eliminate that variable.

Second, look for which coefficients have a simple least common multiple. Eliminate that variable.

If the variables have the same coefficient, you can subtract the two equations to eliminate the variable.

If the variables have opposite coefficients, you add the two equations to eliminate the variable.

Sometimes, you need to multiply one, or both, equations by a number in order to create a common coefficient.

5x + y = 44-4x - y = -34

Solve by Elimination - Click on the terms to eliminate and they will disappear, then add the two equations

together.

3x + y = 15-3x -3y = -21

Solve by Elimination - Click on the terms and they will disappear then add the two equations together.

5x + y = 17-2x + y = -4

Solve by Elimination - There are 2 ways to complete this problem. See both examples.

5x + y = 17 2x - y = 4

( )- 7x = 21

One method is to recognize that the y-coefficient is the same. You can multiply the second equation by -1. This will create opposite coefficients for the y variable. Then, add the two equations.

5x + y = 17-2x + y = -4

Solve the system by elimination.

 4x + 3y = 16 2x - 3y = 8

Pull  4x + 3y = 16 4(4) + 3y = 16

+ 2x - 3y = 8  16 + 3y = 16 6x = 24    3y = 0  x = 4  y = 0 (4, 0)

A (5,1)

B (-5,-1)

C (1,5)

D no solution

Solve each system by elimination.x + y = 6x - y = 4

21 Solve each system by elimination.2x + y = -52x - y = -3

A (-2,1)

B (-1,-2)

C (-2,-1)

D infinitely many

22 Solve each system by elimination.2x + y = -63x + y = -10

A (4,2)

B (3,5)

C (2,4)

D (-4,2)

23 Solve each system by elimination.4x - y = 5x - y = -7

A no solution

B (4,11)

C (-4,-11)

D (11,-4)

24 Solve each system by elimination.3x + 6y = 48-5x + 6y = 32

A (2,-7)

B (7,2)

C (2,7)

D infinitely many

Sometimes, it is not possible to eliminate a variable by adding or subtracting the equations.

When this is the case, you need to multiply one or both equations by a nonzero number in order to create a common coefficient. Then add or subtract the equations.

Examine each system of equations.Which variable would you choose to eliminate?What do you need to multiply each equation by?

2x + 5y = -1 x + 2y = 0

3x + 8y = 815x - 6y = -39

3x + 6y = 62x - 3y = 4

In order to eliminate the y, you need to multiply first.

 3x + 4y = -10 5x - 2y = 18

Multiply the second equation by 2 so the coefficients are opposites. 2(5x - 2y = 18) 10x - 4y = 36

Now solve by adding the equations together. 3x + 4y = -10 10x - 4y = 36 13x = 26  x = 2

Solve for y, by substituting x = 2 into one of the equations.  3x + 4y = -10 3(2) + 4y = -10  6 + 4y = -10  4y = -16  y = -4

So (2,-4) is the solution.

Check:   3x + 4y = -10   5x - 2y = 183(2) + 4(-4) = -10    5(2) - 2(-4) = 18  6 + -16 = -10  10 + 8 = 18 -10 = -10  18 = 18

Now solve the same system by eliminating x. What do you multiply the two equations by?

 3x + 4y = -10 5x - 2y = 18

Multiply the first equation by 5 and the second equation by 3 so the coefficients will be the same 5(3x + 4y = -10) 3(5x - 2y = 18)  15x + 20y = -50   15x - 6y = 54

Now solve by subtracting the equations. 15x + 20y = -50  15x - 6y = 54 26y = -104  y = -4

Solve for x, by substituting y = -4 into one of the equations.  3x + 4y = -10 3x + 4(-4) = -10  3x + -16 = -10  3x = 6    x = 2

So (2,-4) is the solution. Check:   3x + 4y = -10   5x - 2y = 183(2) + 4(-4) = -10    5(2) - 2(-4) = 18  6 + -16 = -10  10 + 8 = 18 -10 = -10  18 = 18

25 Which variable can you eliminate with the least amount of work?

9x + 6y = 15-4x + y = 3

3x - 7y = -2-6x + 15y = 9

x - 3y = -72x + 6y = 34

28 What will you multiply the first equation by in order to solve this system using elimination?

2x + 5y = 203x - 10y = 37

Now solve it.... (11, ) 25

click for answer

(-5,-2)

3x + 2y = -19x - 12y = 19

Now solve it....

click for answer

x + 3y = 43x + 4y = 2

Now solve it.... (-2,2)

click for answer

Choose Your Strategy

Altogether 292 tickets were sold for a basketball game. An adult ticket costs $3. A student ticket costs $1. Ticket sales were $470.

Let a = adults s = students

Set up the system:

 number of tickets sold: a + s = 292 money collected:    3a + s = 470

First eliminate one variable.  a + s = 292 - in both equations s has the same 3a + s = 470 coefficient so you subtract the 2 -2a+ 0 = -178  equations in order to eliminate it. a = 89 -solve for a

Then, find the value of the eliminated variable. a + s = 29289 + s = 292 -substitute 89 for a in 1st equation s = 203 -solve for s

There were 89 adult tickets and 203 student tickets sold.

(89, 203)

Check: a + s = 292  3a + s = 47089 + 203 = 292 3(89) + 203 = 470 292 = 292   267 + 203 = 470  470 = 470

31 A piece of glass with an initial temperature of 99 F is cooled at a rate of 3.5 F/min. At the same time, a piece of copper with an initial temperature of 0 F is heated at a rate of 2.5 F/min. Let m = the number of minutes and t = the temperature in F. Which system models the given information?

t = 99 + 3.5mt = 0 + 2.5m

t = 99 - 3.5mt = 0 + 2.5m

t = 99 + 3.5mt = 0 - 2.5m

32 Which method would you use to solve the system?

A graphing

B substitution

C elimination

t = 99 - 3.5mt = 0 + 2.5m

Now solve it...m = 16.5 t = 41.25

This means that in 16.5 minutes, the temperatures will both be 41.25℃.

click for answer

click for equations

33 What method would you choose to solve the system?

A graphing

B substitution

C elimination

4s - 3t = 8t = -2s -1

D (-2, )

34 Now solve the system!

A ( , -2) 4s - 3t = 8t = -2s -1

B ( , 2)

C (2 , -2)

A graphing

B substitution

C elimination

y = 3x - 1y = 4x

36 Now solve it!

A (1, 4)

B (-4, -1)

C (-1, 4)

y = 3x - 1y = 4x

D (-1, -4)

A graphing

B substitution

C elimination

3m - 4n = 13m - 2n = -1

38 Now solve it!

A (-2, -1)

B (-1, -1)

C (-1, 1)

3m - 4n = 13m - 2n = -1

D (1, 1)

A graphing

B substitution

C elimination

y = -2xy = -0.5x + 3

40 Now solve it!

A (-6, 12)

B (2, -4)

y = -2xy = -0.5x + 3

C (-2, 4)

D (1, -2)

A graphing

B substitution

C elimination

2x - y = 4x + 3y = 16

42 Now solve it!

A (6, 5)

B (-4, 7)

C (-4, 4)

2x - y = 4x + 3y = 16

D (4, 4)

A graphing

B substitution

C elimination

u = 4v3u - 3v = 7

44 Now solve it!

A ( , )

B ( , )

C (28, 7)

u = 4v3u - 3v = 7

D (7, ) 7 4

45 Choose a strategy and then answer the question.What is the value of the y-coordinate of the solution to the system of equations x − 2y = 1 and x + 4y = 7?

ModelingSituations

A group of 148 people is spending five days at a summer camp. The cook ordered 12 pounds of food for each adult and 9 pounds of food for each child. A total of 1,410 pounds of food was ordered.

Part A: Write an equation or a system of equations that describes the above situation and define your variables.

Part B: Using your work from part A, find:

(1) the total number of adults in the group

(2) the total number of children in the group

Tanisha and Rachel had lunch at the mall. Tanisha ordered three slices of pizza and two colas. Rachel ordered two slices of pizza and three colas. Tanisha’s bill was $6.00, and Rachel’s bill was $5.25. What was the price of one slice of pizza? What was the price of one cola?

Sharu has $2.35 in nickels and dimes. If he has a total of thirty-two coins, how many of each coin does he have?

Ben had twice as many nickels as dimes. Altogether, Ben had $4.20.How many nickels and how many dimes did Ben have?

46 Your class receives $1105 for selling 205 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9

Set up a system and solve. How many packages of cards were sold?

You will answer how many packages of gift wrap in the

next question.

47 Your class receives $1105 for selling 205 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9

Set up a system and solve. How many packages of gift wrap were sold? S

48 The sum of two numbers is 47, and their difference is 15. What is the larger number?

49 Ramon rented a sprayer and a generator. On his first job, he used each piece of equipment for 6 hours at a total cost of $90. On his second job, he used the sprayer for 4 hours and the generator for 8 hours at a total cost of $100. What was the hourly cost for the sprayer?

50 What is true of the graphs of the two lines 3y - 8 = -5x and 3x = 2y -18?

A no intersection

B intersect at (2,-6)

C intersect at (-2,6)

D are identical

51 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. Set up a system to solve. Which method will you use? (Solving it comes later...)

A graphing

B substitution

C elimination

52 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many quarters do you have?

53 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many nickels do you have?

54 Julia went to the movies and bought one jumbo popcorn and two chocolate chip cookies for $5.00. Marvin went to the same movie and bought one jumbo popcorn and four chocolate chip cookies for $6.00. How much does one chocolate chip cookie cost?

A $0.50

B $0.75

C $1.00

D $2.00

55 Mary and Amy had a total of 20 yards of material from which to make costumes. Mary used three times more material to make her costume than Amy used, and 2 yards of material was not used. How many yards of material did Amy use for her costume?

56 The tickets for a dance recital cost $5.00 for adults and $2.00 for children. If the total number of tickets sold was 295 and the total amount collected was $1220, how many adult tickets were sold?

Solving Systems of Inequalities

Two or more linear inequalities form a system of inequalities.

A solution to the system is an ordered pair that is a solution of each inequality in the system.

Since inequalities have more than one solution, the solutions are best shown in a graph.

Graphing a System of Linear Inequalities

1. Graph the boundary lines of each inequality. (Remember use a dashed line for < and >   and a solid line for < and >)

2. Shade the half-plane for each inequality.

3. The intersection of the half-planes is the solution.

Solve the system of inequalities.

x + 2y < 6-x + y < 0

Pull Pull

2x + y > -4 x - 2y < 4

Pull Pull

4x + 2y < 84x + 2y > -8

Pull Pull

Graph the following system of inequalities on the set of axes shown below and label the solution set S.

y > −x + 2

y ≤ 2 x + 5

A company manufactures bicycles and skateboards. The company’s daily production of bicycles cannot exceed 10, and its daily production of skateboards must be less than or equal to 12. The combined number of bicycles and skateboards cannot be more than 16. If x is the number of bicycles and y is the number of skateboards, graph on the accompanying set of axes the region that contains the number of bicycles and skateboards the company can manufacture daily.

Write a system of inequalities from the graph.

57 Solve the system of linear inequalities.

y > -2x + 1y < x + 2

x > 2y < 5

-2x - 2y < 4y - 2x > 1

-5x + y > -24x + y < 1

3x + 2y < 122x - 2y < 20 S

A (0,4)

B (2,4)

C (-4,1)

D (4,-1)

Which point is in the solution set of the system of inequalities shown in the accompanying graph?

63 Which ordered pair is in the solution set of the system of inequalities shown in the accompanying graph?

A (0, 0)

B (0, 1)

C (1, 5)

D (3, 2)

64 Which ordered pair is in the solution set of the following system of linear inequalities?

A (0,3)

B (2,0)

C (−1,0)

D (−1,−4)

y < 2x + 2y ≥ −x − 1

65 Mr. Braun has $75.00 to spend on pizzas and soda for a picnic. Pizzas cost $9.00 each and the drinks cost $0.75 each. Five times as many drinks as pizzas are needed. What is the maximum number of pizzas that Mr. Braun can buy?

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