Substitution - Rochester City School District

[Pages:11]Algebra/Geometry Blend

Name

Unit #4: Systems of Linear Equations & Inequalities

Lesson 3: Solving Systems of Linear Equations Algebraically

Period Date

[DAY #1]

Earlier we talked about how to find the solution to a system of equations by graphing each equation. That is often time a LOT more work and not always the best choice. Who walks around with graph paper or a graphing calculator?

You can also solve equations algebraically. In fact, there are 2 different methods for solving equations algebraically. Today we are going to learn and explore how to solve systems of equations using substitution.

Substitution

To substitute is to

a variable with something of equal value.

Examples: Find the solution to the following system of equations algebraically

#1. y = 2x 6x ? y = 8

Steps 1) Substitute..... plug in the equation that has

the variable __________ into the other equation.

2) Solve the equation for the first ______________

3) Substitute...... plug in the answer for the variable that you just found into one of the ________________ equations and ____________ for the other variable.

4) Write out your answer.

#2.

3x ? 5y = 11

x = 3y + 1

#3.

y = x ? 4

2x + y = 5

#4.

x + 4y = 6

x = -y + 3

#5.

2x + y = 1

x = 23 + 4y

[DAY #2]

Solve the following systems of equations by substitution.

1.)

3x + 4y = 2

2.)

x = y + 3

2x ? 5y = 14 y = 3x + 5

3.)

3x ? y = 12

y = 2x ? 7

4.)

5x + y = 9

3x + 2y = -3

Try these...

1.)

3x + 2y = 16

x = 2y ? 8

2.)

3x ? 8y = 17

y = 2x ? 7

3.)

5x ? y = 8

y = 3x

4.)

3x + y = 13

5x + 4y = -4

[DAY #3] In your own words, write what you think the word Elimination means?

Example #1: Solve this system of linear equations using the elimination method.

Let's solve this system of equations by eliminating the `x' variable. If you look at the `x'

terms, we have an `x' and a `2x'. What will cancel out the `2x' term?

.

We need to multiply the `x' term by

to turn it into

so it will

cancel out with the `2x' term. But when we do, we need to multiply EVERYTHING in that

equation by the same number.

ORIGINAL SYSTEM 2 + = 6 - 3 = -11

NEW SYSTEM

SOLUTION

Once you have the NEW SYSTEM, add the two equations together. One of the variables should cancel out. Solve the remaining equation for the variable. Substitute that variable into EITHER of the original equations (YOUR CHOICE!!!) and solve. Write out your answer!

Now let's solve this system of equations by eliminating the `y' variable. If you look at the `y'

terms, we have an `y' and a `-3y'. What will cancel out the `-3y' term?

.

We need to multiply the `y' term by

to turn it into

so it will cancel out

with the `-3y' term. But when we do, we need to multiply EVERYTHING in that equation by

the same number.

ORIGINAL SYSTEM 2 + = 6 - 3 = -11

NEW SYSTEM

SOLUTION

 When using Elimination you must _____________________________ the entire equation by a constant to eliminate one of the

Sometimes you might need to multiply BOTH equations by a to eliminate the variable

Example #2: Solve this system of linear equations using the elimination method.

ORIGINAL SYSTEM 2 + 3 = 7 - = 1

NEW SYSTEM

SOLUTION

Example #3: Solve this system of linear equations using the elimination method.

ORIGINAL SYSTEM 2x ? 5y = 18 7x + y = 26

NEW SYSTEM

SOLUTION

Try these...

Solve each system of equations by writing a new system that eliminates one of the

variables.

1.)

5x + y = 6

2.)

2x + 5y = -20

3x ? 4y = 22

7x + 5y = 5

3.)

7x ? 3y = 2

2x + 6y = - 20

4.)

9x + 5y = 5

2x + 10y = -30

[DAY #4]

Some times when solving a system of equations by elimination, you have to multiply

BOTH equations in order for something to cancel out.

1.)

3x ? 5y = 29

2.)

8x ? 3y = - 13

2x + 3y = -6

3x + 5y = - 11

3.)

3x + 4y = -10

2x + 5y = -2

4.)

6x ? 4y = 38

5x ? 3y = 31

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