Solving Linear Systems by Substitution

Section 3-2:

Solving a Linear System by SubstitutionStep 1: Solve one equation for one of

its variables.Step 2: Substitute the expression from

step 1 into the other equation and solve for the other variable.

Step 3: Substitute the value from Step 2 into the revised equation from Step 1 and solve.

Step 4: Check the solution in each of the original equations.

Use SubstitutionExample 1

Solve the system using substitution.

6=y–4x

2x=y

SOLUTION

Substitute 2x for y in Equation 2. Solve for x.

Write Equation 2.6=y–4x

Substitute 2x for y.6=2x–4x

6=2x Combine like terms.

3=x Solve for x.

Equation 1

Equation 2

Use SubstitutionExample 1

Substitute 3 for x in Equation 1. Solve for y.

Write Equation 1.2x=y

Substitute 3 for x.( )32=y

Solve for y.6=y

You can check your answer by substituting 3 for x and 6 for y in both equations.

ANSWER

The solution is .( )3, 6

Use SubstitutionExample 2

Solve the system using substitution.

Equation 17=2y3x +

Equation 23=2y–x –

SOLUTION

STEP 1 Solve Equation 2 for x.

Choose Equation 2 because the coefficient of x is 1.3=2y–x –

Solve for x to get revised Equation 2.3=x –2y

Use SubstitutionExample 2

STEP 2 Substitute 2y 3 for x in Equation 1. Solve for y.–

Write Equation 1.7=2y3x +

7=2y+96y – Use the distributive property.

7=98y – Combine like terms.

16=8y Add 9 to each side.

2=y Solve for y.

7=2y3 +( )32y – Substitute 2y 3 for x.–

Use SubstitutionExample 2

STEP 3 Substitute 2 for y in revised Equation 2. Solve for x.

Write revised Equation 2.3= 2yx –

Substitute 2 for y.3= 2x –( )2

Simplify.= 1x

STEP 4 Check by substituting 1 for x and 2 for y in the original equations.

Equation 1 Equation 2

7=2y3x + =2y–x 3–Write original equations.

Use SubstitutionExample 2

73( )1 + 2( )2 =? 1 2( )2 =

?– 3–Substitute for x and y.

73 + 4 =? 1 4 =

?– 3–Simplify.

77 = = 3–3–Solution checks.

ANSWER

The solution is . ( )1, 2

Solve the system using substitution. Tell which equation you chose to solve and use for the substitution. Explain.

Checkpoint

1.

Use Substitution

3=y2x +

0=y3x +

ANSWER

Sample answer: The second equation; this equation had 0 on one side and the coefficient of y was 1, so I solved for y to obtain y 3x.= –

( ), 93– .

Solve the system using substitution. Tell which equation you chose to solve and use for the substitution. Explain.

Checkpoint

2.

Use Substitution

4=3y2x +

1=2yx +

ANSWER

Sample answer: The second equation; the coefficient of x in this equation was 1, so solving for x gave a result that did not involve any fractions.

( )5, 2– .

Solve the system using substitution. Tell which equation you chose to solve and use for the substitution. Explain.

Checkpoint

3.

Use Substitution

10=2y4x +

5=y3x –

ANSWER

Sample answer: The first equation; the coefficient of y in this equation was 1, so solving for y gave a result that did not involve any fractions.

–( )2, 1 .

Homework:

p. 135 17-22 all

Write and Use a Linear SystemExample 3

Museum Admissions On one day, the Henry Ford Museum in Dearborn, Michigan, admitted 4400 adults and students and collected $57,200 in ticket sales. The price of admission is $14 for an adult and $10 for a student. How many adults and how many students were admitted to the museum that day?

SOLUTION

VERBALMODEL

Total number

admitted Number

of adults Number

of students =+

• =+Total amount

collected •

Studentprice

Adultprice

Numberof adults

Numberof students

Write and Use a Linear SystemExample 3

LABELS Number of adults x = (adults)

Number of students y = (students)

Total number admitted 4400 =

(dollars) Price for one adult 14=

Price for one student 10 =

(people)

Total amount collected 57,200 =

(dollars)

(dollars)

ALGEBRAICMODEL

Equation 1 (number admitted) 4400=x + y

Equation 2 (amount collected) 57,200=14x + 10y

Write and Use a Linear SystemExample 3

Use substitution to solve the linear system.

4400=x y– Solve Equation 1 for x; revised Equation 1.

57,200=+ 10y14y 61,600 – Use the distributive property.

57,200=4y61,600 – Combine like terms.

57,200=14 + 10y( )y4400 – Substitute 4400 y for x in Equation 2.

–

4400=4y– – Subtract 61,600 from each side.

1100=y Divide each side by 4. –

Write and Use a Linear SystemExample 3

ANSWER

There were 3300 adults and 1100 students admitted to the Henry Ford Museum that day.

4400=x y– Write revised Equation 1.

4400=x 1100 – Substitute 1100 for y.

3300=x Simplify.

Checkpoint Write and Use a Linear System

4. On another day, the Henry Ford Museum admitted 1300 more adults than students and collected $56,000. How many adults and how many students were admitted to the museum that day?

ANSWER 2875 adults and 1575 students