solving linear systems by substitution section 3-2:
Post on 01-Jan-2016
227 Views
Preview:
TRANSCRIPT
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
top related