Warm-Up Solving Systems of Linear Equations: Substitution - Edgenuity Inc.

[Pages:12]Warm-Up

Solving Systems of Linear Equations: Substitution

?

Lesson For a system of two equations, how can substituting for

Question one variable help you solve for the other?

Lesson Goals

Create a system

of linear equations from a graph or table in slopeintercept form.

Isolate a variable

in one of the equations of a linear system.

Solve a system of linear equations using

substitution .

W2K

Words to Know

Write the letter of the definition next to the matching word as you work through the lesson. You may use the glossary to help you.

C interpret

A. to take the place of; to replace

B systems of linear

equations

A substitute

B. a set of linear equations that have the same variables; has one solution if the lines intersect, infinitely many solutions if the lines are the same, and no solution if the lines are parallel

C. to explain in understandable terms; to understand according to personal beliefs

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Warm-Up

Solving Systems of Linear Equations: Substitution

W2K

Words to Know

E

linear equation

D isolate

D. to separate from other substances; to place apart so as to be alone.

E. an algebraic equation with constants and variable terms of highest degree 1

Graphing a System to Verify the Solution

Solve the system of linear equations.

=

-

1 4

-

2

= 1

Substitute the value of = 1 into the equation.

1 = - 4 ( 1 ) - 2

1 = - 4 - 2

9 = - 4 =

-2.25

The solution is (1, -2.25 ).

(1, ? )

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Instruction

Solving Systems of Linear Equations: Substitution

Slide

2

Writing a System of Equations from a Graph

Example: Marcel starts running and gets a

3-mile head start on Jonah, who is riding

Distance (miles)

his bike. The graphs show their travel.

When does Jonah catch up to Marcel?

Put each line into slope-intercept form.

Marcel:

4-3 1 = 10 - 0 = 10 = 0.1

= 0.1 + 3

(10, 4) Marcel

(0, 3)

(10, 3)

Jonah

(0, 0)

Time (minutes)

Jonah:

3-0 3 = 10 - 0 = 10 = 0.3

= 0.3

How to Solve a System of Equations Using Substitution

1. Isolate one variable in the system of linear equations, if needed.

2. Use substitution to create a one-variable linear equation. 3. Solve to determine the unknown variable in the equation.

4. Substitute the value of the variable into either original equation to solve

for the other variable. 5. Write and interpret the solution to the system of equations. 6. Check the solution.

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Instruction

Solving Systems of Linear Equations: Substitution

Slide

2

Using Substitution to Solve a System of Equations

Marcel starts running and gets a 3-mile head start on Jonah, who is riding his bike.

The graphs show their travel. When does Jonah catch up to Marcel?

1. Isolate one variable in the system of linear equations, if needed.

2. Use substitution to create a onevariable linear equation.

3. Solve to determine the unknown variable in the equation.

( 0.3

= 0.1 + 3 = 0.3

) = 0.1 + 3

-0.1 - 0.1 0.2 3 0.2 = 0.2

= 15

Marcel starts running and gets a 3-mile head start on Jonah, who is riding his bike. The graphs show their travel. When does Jonah catch up to Marcel?

1. Isolate one variable, if needed. 2. Use substitution to create a one-

variable linear equation. 3. Solve for . 4. Substitute the value of into either

original equation to solve for .

5. Write and interpret the

solution to the system of equations.

= 0.1 + 3 = 0.3 = 15

= 0.1 15

= 1.5 + 3

= 4.5

+3

After 15 minutes that Jonah

has been on his bike, he meets up

with Marcel 4.5 miles away.

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Instruction

Solving Systems of Linear Equations: Substitution

Slide

2

Verify the Solution from a Graph

Marcel starts running and gets a 3-mile

head start on Jonah, who is riding his

bike. The graphs show their travel. When

does Jonah catch up to Marcel?

6. Check the solution (15, 4.5).

Jonah is going to catch up with Marcel

15 minutes after they leave at 4.5 miles away from the park.

Distance (miles)

Marcel Jonah

Time (minutes)

5

Writing a System of Equations from Tables

Write and solve a system of linear equations from tables.

+1

0

1

2

3

6

+5.5 +1

0

11.5

1

17

2

22.5

3

4 10.75 17.5 24.25

+6.75

= 5.5

= 5.5 + 6

= 6.75 = 6.75 + 4

We know from the tables that the intersection is somewhere between = 1 and = 2.

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Instruction

Solving Systems of Linear Equations: Substitution

Slide

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Using Substitution to Solve a System of Equations

Solve the system of linear equations represented by the tables.

0

6

1

11.5

2

17

3

22.5

= 5.5 + 6

0

4

1

10.75

2

17.5

3

24.25

= 6.75 + 4

5.5 + 6 = 6.75 + 4 -5.5 - 4 - 5.5 - 4

2 1.25 1.25 = 1.25

1.6 =

Substitute the value of into either original equation to get the value of .

= 5.5 1.6

= 8.8 + 6 = 14.8

+6

The solution is (1.6, 14.8 ) .

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Instruction

Solving Systems of Linear Equations: Substitution

Slide

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Finding the Number of Raffle Tickets Sold

As people entered the gym for a volleyball tournament, members of the pep club

were selling $3 raffle tickets for a tablet and $4 raffle tickets for a gaming system.

The number of $4 tickets sold was four more than one half the number of $3

tickets sold. The pep club raised $636 from the ticket sales. Write a system of

equations to represent the given scenario.

= # of $ 3 tickets

3 + 4 = 636

= # of $ 4 tickets

1 = 2 + 4

Solving a System Using Substitution

1. Isolate one variable in the system of linear equations, if needed.

2. Use substitution to

create one-variable linear equation. 3. Solve to determine the unknown

variable in the equation. 4. Substitute the value of the variable

into either original equation to solve for the other variable.

Solve: 3 + 4 = 636

1 = 2 + 4 1 3 + 4 2 + 4 = 636 3 + 2 + 16 = 636

5 + 16 = 636

-16 -16 5 620 5= 5

= 124124

1 = 2 ( 124

= 62 + 4

)+4

= 66

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Instruction

Solving Systems of Linear Equations: Substitution

Slide

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Interpreting and Checking the Solution

How many $3 raffle tickets and how many $4 raffle tickets were sold?

= # of $3 raffle tickets = # of $4 tickets Solution: (, )

3 + 4 = 636 1

= 2 + 4

5. Interpret the solution to the system of equations.

124 $3 raffle tickets were sold.

66 $4 raffle tickets were sold.

6. Check the solution.

3( 124 ) + 4( 66 )

372 + 264 = 636

1 66 = 2 124 + 4 66 = 62 + 4

66 = 66

10

Solving a System Using the Substitution Method

Solve the system of equations using substitution. 6 - 4 = -15 + 4 = 1 Solve the second equation for .

= 1 - 4

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