Transcript Solving Systems of Equations by Elimination PowerPoint

Solving Systems of Equations by
Elimination (6-3, 6-4)
Objective: Solve systems of equations by
using elimination with addition, subtraction,
and multiplication. Solve real-world
problems involving systems of equations.
Elimination Using Addition
Using addition or subtraction to solve a system is
called elimination.
Use the following steps to solve a system by
elimination:
1.
2.
3.
Write the system so like terms with the same or
opposite coefficients are aligned.
Add or subtract the equations, eliminating one variable.
Then solve the equation.
Substitute the value from Step 2 into one of the
equations and solve for the other variable. Write the
solution as an ordered pair (x, y).
Example 1
Use elimination to solve the system of equations.
-3x + 4y = 12
3x – 6y = 18
-3x + 4y = 12
+3x – 6y = 18
-2y = 30
y = -15
(-24, -15)
-3x + 4(-15) = 12
-3x – 60 = 12
+60 +60
-3x = 72
x = -24
Check Your Progress
Choose the best answer for the following.
Use elimination to solve the system of equations.
3x – 5y = 1
3(2) – 5y = 1
3x – 5y = 1
2x + 5y = 9
6 – 5y = 1
2x + 5y = 9
-6
-6
A. (1, 2)
5x = 10
-5y = -5
B. (2, 1)
x=2
y=1
C. (0, 0)
D. (2, 2)
Solving Systems - Elimination
We can use elimination to find specific numbers that are
described as being related to each other.
Example 2
Four times one number minus three times
another number is 12. Two times the first
number added to three times the second number
is 6. Find the numbers.
4x – 3y = 12
2x + 3y = 6
6x = 18
x=3
4(3) – 3y = 12
12 – 3y = 12
-12
-12
-3y = 0
y=0
The numbers
are 3 and 0.
Check Your Progress
Choose the best answer for the following.
Four times one number added to another number is
-10. Three times the first number minus the second
number is -11. Find the numbers.
A.
B.
C.
D.
-3, 2
-5, -5
-5, -6
1, 1
4x + y = -10
3x – y = -11
7x = -21
x = -3
4(-3) + y = -10
-12 + y = -10
+12
+12
y=2
Elimination Using Subtraction
Sometimes we can eliminate a variable by
subtracting one equation from another.
Example 3
Use elimination to solve the system of equations.
4x + 2y = 28
4x – 3y = 18
5y = 10
y=2
4x + 2(2) = 28
4x + 4 = 28
-4 -4
4x = 24
x=6
(6, 2)
Check Your Progress
Choose the best answer for the following.
Use elimination to solve the system of equations.
9x – 2y = 30
9(2) – 2y = 30
9x – 2y = 30
x – 2y = 14
18 – 2y = 30
x – 2y = 14
A.
B.
C.
D.
(2, 2)
(-6, -6)
(-6, 2)
(2, -6)
8x = 16
x=2
-18
-18
-2y = 12
y = -6
Example 4
A hardware store earned $956.50 from renting ladders and
power tools last week. The store charged customers for a total
of 36 days for ladders and 85 days for power tools. This week
the store charged 36 days for ladders, and 70 days for power
tools, and earned $829. How much does the store charge per
day for ladders and for power tools?
$6.50 per day
36x + 85y = 956.50 36x + 85(8.50) = 956.50
for ladders
36x + 722.50 = 956.50 and $8.50 per
36x + 70y = 829
-722.50 -722.50 day for power
15y = 127.50
36x = 236
tools.
y = 8.50
x = 6.50
Check Your Progress
11(21) + 13y = 523.50
231 + 13y = 523.50
13y = 292.50
y = 22.50
Choose the best answer for the following.
For a school fundraiser, Marcus and Anisa participated in
a walk-a-thon. In the morning, Marcus walked 11 miles
and Anisa walked 13. Together they raised $523.50.
After lunch, Marcus walked 14 miles and Anisa walked
13. In the afternoon they raised $586.50. How much did
each raise per mile of the walk-a-thon?
A.
B.
C.
D.
Marcus:
Marcus:
Marcus:
Marcus:
$22.00, Anisa:
$21.00, Anisa:
$24.00, Anisa:
$20.75, Anisa:
$21.65
$22.50
$20.00
$22.75
11x + 13y = 523.50
14x + 13y = 586.50
-3x = -63
x = 21
Elimination Using Multiplication
Sometimes neither variable can be eliminated by adding or
subtracting.
You can use multiplication to solve.
Use the following steps to solve a problem by elimination that
requires multiplication:
1.
2.
3.
Multiply at least one equation by a constant to get two equations
that contain opposite terms.
Add the equations, eliminating one variable. Then solve the
equation.
Substitute the value from Step 2 into one of the equations and
solve for the other variable. Write the solution as an ordered pair
(x, y).
Example 5
Use elimination to solve the system of equations.
-2( 2x + y = 23 )  -4x – 2y = -46
3x + 2y = 37  3x + 2y = 37
-x = -9
x=9
(9, 5)
2(9) + y = 23
18 + y = 23
-18
-18
y=5
Check Your Progress
Choose the best answer for the following.
Use elimination to solve the system of equations.
-3( x + 7y = 12 )  -3x – 21y = -36
x + 7(1) = 12
x + 7 = 12
3x – 5y = 10  3x – 5y = 10
-26y = -26
-7 -7
A. (1, 5)
x=5
y=1
B. (5, 1)
C. (5, 5)
D. (1, 1)
Elimination Method
Sometimes you have to multiply each equation
by a different number in order to solve the
system.
Example 6
Use elimination to solve the system of equation.
5(4x + 3y = 8 )  20x + 15y = 40
3(3x – 5y = -23)  9x – 15y = -69
29x = -29
x = -1
(-1, 4)
4(-1) + 3y = 8
-4 + 3y = 8
+4
+4
3y = 12
y=4
Check Your Progress
Choose the best answer for the following.
Use elimination to solve the system of equations.
2(3x + 2y = 10 )  6x + 4y = 20 3x + 2(-1) = 10
3x – 2 = 10
-3( 2x + 5y = 3 )  -6x – 15y = -9
+2
+2
-11y
=
11
A. (-4, 1)
3x = 12
y
=
-1
B. (-1, 4)
x=4
C. (4, -1)
D. (-4, -1)
Solve Real-World Problems
Sometimes it is necessary to use multiplication
before elimination in real-world problem solving
too.
Example 7
A fishing boat travels 10 miles downstream in 30
minutes. The return trip takes the boat 40 minutes.
Find the rate in miles per hour of the boat in still
water.
x: rate of boat in still water.
y: rate of current
30 minutes = ½ hour
40 minutes = 2/3 hour
2(½ (x + y) = 10 )
3/ (2
2 /3 (x – y) = 10 )
x + y = 20
x – y = 15
2x = 35
x = 17.5 mph
Check Your Progress
Choose the best answer for the following.
A helicopter travels 360 miles with the wind in 3
hours. The return trip against the wind takes the
helicopter 4 hours. Find the rate of the helicopter in
still air.
3(x + y) = 360
x + y = 120
A. 103 mph
4(x – y) = 360
x – y = 90
B. 105 mph
2x = 210
C. 100 mph
D. 17.5 mph