Holt Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems 3-1,2 Solving Linear Systems Holt Algebra 2 Vocab (Slide #5–8) Vocab (Slide #5–8) Lesson

Holt Algebra 2

3-1 Using Graphs and Tables to Solve Linear Systems

3-1,2 Solving Linear Systems

Holt Algebra 2

Vocab (Slide #5–8) Vocab (Slide #5–8)

Lesson Presentation (Slide #9–31)Lesson Presentation (Slide #9–31)

Lesson Quiz (Slide #33-34)Lesson Quiz (Slide #33-34)

Text Questions (NONE) Text Questions (NONE)

Objective and Standards (Slide #4)Objective and Standards (Slide #4)

Warm Up (Slide #2-3)Warm Up (Slide #2-3)

Worksheets 3.1A, 3.2A (Slide #32)Worksheets 3.1A, 3.2A (Slide #32)

Holt Algebra 2


Warm UpUse substitution to determine if (1, –2) is an element of the solution set of the linear equation.

1. y = 2x + 1 2. y = 3x – 5 no yes

Write each equation in slope-intercept form.3. 2y + 8x = 6 4. 4y – 3x = 8

y = –4x + 3


Holt Algebra 2


Warm UpDetermine if the given ordered pair is an element of the solution set of

2x – y = 5

3y + x = 6

1. (3, 1) yes 2. (–1, 1) no

Solve each equation for y.

3. x + 3y = 2x + 4y – 4

4. 6x + 5 + y = 3y + 2x – 1

y = –x + 4

y = 2x + 3


Holt Algebra 2


1. Solve systems of linear equations with:•Graphs and tables•Substitution•Elimination

2. Determine whether there will be one, none, or an infinite number of solutions by noting characteristics of each equation.

Objectives


Holt Algebra 2


system of equationslinear systemsubstitutioneliminationlinear combinations

Vocabulary


Holt Algebra 2


A system of equations is a set of two or more equations containing two or more variables. A linear system is a system of equations containing only linear equations.

A line is an infinite set of points that are solutions to a linear equation. The solution of a system of equations is the set of all points that satisfy each equation.


Holt Algebra 2

3-2Using Algebraic Methods to Solve Linear Systems

There are two aspects of substitution:

In one, a possible solution (ordered pair) is given and you simply substitute its x and y values into each equation to see if that point satisfies both.

In the other, you substitute the equivalent expression for a variable from one equation into the other equation, solve for one variable, then use that value to solve for the other variable.

(I know…it sounds all so confusing, but it’s really easy.)


Holt Algebra 2


You can also solve systems of equations with the elimination method.

With elimination, you get rid of one of the variables by adding or subtracting equations.

You may have to multiply one or both equations by a number to create variable terms that can be eliminated.

The elimination method is sometimes called the addition method or linear combinations.

Reading Math


Holt Algebra 2


Points to remember about linear equations and systems:

On the graph of the system of two equations, the solution

is the set of points where the lines intersect.

A point is a solution to a system of equation if the x- and

y-values of the point satisfy both equations.


Holt Algebra 2


To see if a given point is a solution to a linear system, substitute the (x,y) values into both equations.

For example… “Is (1,3) the solution to this linear system?”

Example 1A: Verifying Solutions of Linear Systems

(1, 3); x – 3y = –8

3x + 2y = 9


Ans: YES

Holt Algebra 2


Check It Out! Example 1b

Is (5,3) an element of the solution set for the system of equations?

(5, 3); 6x – 7y = 1

3x + 7y = 5


Ans: NO

Holt Algebra 2


Solve the system. Check your answer.

Example 2A: Solving Linear Systems by Using Graphs and Tables on your graphing calculator

2x – 3y = 3

y + 2 = x

First, solve each equation for y, then graph both:


Sol’n: (3,1)

On the graph, the lines appear to intersect at the ordered pair (3, 1)Use the calculator’s “Table” or “Trace”function to verify.

Holt Algebra 2


Use a graph and a table to solve the system. Check your answer.

x + y = 8

2x – y = 4


First, solve each equation for y.


Sol’n: (4, 4)

Holt Algebra 2


Different Slopeswill have

ONE solution

Same Slopesand Same Y-int.

will haveINFINITE Sol’ns.

Same Slopesbut Different Y-int.

will haveNO Solutions.


Holt Algebra 2


An identity, such as 0 = 0, 8 = 8, -7 = -7, etc…

is always true and indicates infinite solutions.

A contradiction, such as 1 = 3, 5 = 9, -8 = 8, etc…

is never true and indicates no solution.

Remember!


Holt Algebra 2


One golf course charges $20 to rent golf clubs plus $55 per hour for golf cart rental.

A different course charges $35 to rent clubs plus $45 per hour to rent a cart.

Both places allow rentals in ½hr. increments.

Q: For what number of hours is the cost of renting clubs and a cart the same for each course?

Example 4: Summer Sports Application


Holt Algebra 2


Example 4 Continued

Let x represent the number of hours and y represent the total cost in dollars.

City Park Golf Course: y = 55x + 20

Sea Vista Golf Course: y = 45x + 35

Step 1 Write an equation for the cost of renting clubsand a cart at each golf course.

Sol’n: 1.5hrs


Holt Algebra 2


Check It Out! Example 4

Ravi is comparing the costs of long distance calling cards. To use card A, it costs $0.50 to connect and then $0.05 per minute. To use card B, it costs $0.20 to connect and then $0.08 per minute. For what number of minutes does it cost the same amount to use each card for a single call?

Step 1 Write an equation for the cost for each of the different long distance calling cards.Let x represent the number of minutes and y represent the total cost in dollars.

Card A: y = 0.05x + 0.50 Card B: y = 0.08x + 0.20


Holt Algebra 2



Step 2 Here’s a situation where, since both are equal to “y”, you can set the two equations equal to each other.

Card A: y = 0.05x + 0.50Card B: y = 0.08x + 0.20


.05x + .50 = .08x + .20 -.05x -.05x -.20 -.20 .30 = .03x ÷.03 ÷.03 10 = xSo, both plans are the same for a 10minute call

Holt Algebra 2


Use variable substitution to solve the system:

Example 1A: Solving Linear Systems by Substitution

y = x – 1

x + y = 7

x + y = 7x + (x – 1) = 7

2x – 1 = 7 2x = 8 x = 4

Step 1: Substitute the equivalent expression for “y” from the first equation in place of “y” in the second equation and solve for “x”.


Then…

Holt Algebra 2


Step 2: Substitute the x-value into one of the original equations to solve for y.

Example 1A Continued

y = x – 1

y = (4) – 1

y = 3

The solution is the ordered pair (4, 3).


Holt Algebra 2


Example 1A Continued

Check A graph or table supports your answer.


Holt Algebra 2


Use substitution to solve the systems of equations.

Example 1B: Solving Linear Systems by Substitution

2y + x = 4

3x – 4y = 7

Sol’n: (3, 1/2)


y = 2x – 1

3x + 2y = 26

Sol’n: (4,7)

5x + 6y = –9

2x – 2 = –y

Sol’n: (3,-4)

Holt Algebra 2


Check A graph or table supports your answer.

Check It Out! Example 1a Continued


Holt Algebra 2


Example 2A: Solving Linear Systems by Elimination

3x + 2y = 4

4x – 2y = –18

Sol’n: (–2, 5)


3x + 5y = –16

2x + 3y = –9

Sol’n: (3,-5)

4x + 7y = –25

–12x –7y = 19

Sol’n: (.75, -4)

Holt Algebra 2


Use elimination to solve the system of equations.

5x – 3y = 42

8x + 5y = 28


Sol’n: (6,-4)


Holt Algebra 2


Example 3: Solving Systems with Infinitely Many or No Solutions

3x + y = 1

2y + 6x = –18

Sol’n: NONE


56x + 8y = –32

7x + y = –4

Sol’n: INFINITE

6x + 3y = –12

2x + y = –6

Sol’n: NONE

Holt Algebra 2


A veterinarian needs 60 pounds of dog food that is 15% protein. He will combine a beef mix that is 18% protein with a bacon mix that is 9% protein. How many pounds of each does he need to make the 15% protein mixture?

Example 4: Zoology Application

Let x present the amount of beef mix in the mixture.

Let y present the amount of bacon mix in the mixture.

Write one equation based on the amount of dog food

Write another equation based on the amount of protein


THEN…

Holt Algebra 2


Solve the system.x + y = 60

0.18x +0.09y = 9

Example 4 Continued


Sol’n: (40, 20)

Holt Algebra 2


A coffee blend contains Sumatra beans which cost $5/lb, and Kona beans, which cost $13/lb. If the blend costs $10/lb, how much of each type of coffee is in 50 lb of the blend?

Let x represent the amount of the Sumatra beans in the blend.


Let y represent the amount of the Kona beans in the blend.

Write one equation based on the amount of each bean

Write another equation based on cost of the beans:


THEN…

Holt Algebra 2


Solve the system.x + y = 50

5x + 13y = 500

Check It Out! Example 4 Continued

3-1,2

Solving Linear Systems

Sol’n: (18.75, 31.25)

Holt Algebra 2


3-1,2

Solving Linear Systems

Holt Algebra 2


Lesson Quiz: Part I

Use substitution to determine if the given ordered pair is an element of the solution set of the system of equations.

x + y = 2

y + 2x = 5

x + 3y = –9

y – 2x = 41. (4, –2) 2. (–3, –2)

Solve the system using a table and graph. Check your answer.

3.x + y = 1

3x –2y = 8


Holt Algebra 2


Lesson Quiz: Part IIWhich system has NO solution and which has INFINITE solutions.

4. 5.–4x = 2y – 10

y + 2x = –10

y + 2x = –10

y + 2x = –10

6. Kayak Kottage charges $26 to rent a kayak plus $24 per hour for lessons. Power Paddles charges $12 for rental plus $32 per hour for lessons. Both places allow rentals in 15min (1/4hr) For what number of hours is the cost of equipment and lessons the same for each company?


Documents

Holt Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems 3-1,2 Solving Linear Systems Holt Algebra 2 Vocab (Slide #5–8) Vocab (Slide #5–8) Lesson