Download HW Chap 5 Review WKD

Chapter 5 Review
3
1. Is (−2 , ) a solution to
4
⎧⎪ x − 12y = −11
⎨
−2x + 8y = 10
⎩⎪
!
3⎞
3
⎛
⎜⎝ −2, ⎟⎠ means x = −2 and y =
4
4
3
into both
4
Equation A and Equation B to
see if they work in both equations
plug x = −2 and y =
Equation A ⎧ x − 12y = −11
⎨
Equation B ⎩−2x + 8y = 10
Equation A
(x) − 12(y) = −11
Equation B
− 2(x) + 8(y) = 10
⎛ 3⎞
−2 − 12 ⎜ ⎟ = −11
⎝ 4⎠
⎛ 3⎞
− 2 (−2) + 8 ⎜ ⎟ = 10
⎝ 4⎠
−2 − 9 = −11
−11 = −11
4 + 6 = 10
10 = 10
3⎞
⎛
⎜⎝ −2, ⎟⎠ works in both equations so
4
YES it is a solution.
Math 100 !
Chapter 5 Review WKD
!
© 2016 Eitel
Solve each system of equations by graphing. List your answers as an ordered pair.
Line A ⎧⎪ y = x + 2
3.
⎨
Line B ⎪ y = −2x + 8
⎩
!
The lines intersect at the point ( 3 , 4 )
Line B
(x,y)
Line A
⎧
−2
x+2
Line A ⎪ 2x + 3y = 6 ⇒ y =
5.
!
3
⎨
Line B ⎪
y=4
⎩
The lines intersect at the point ( –3 , 4 )
!
(x,y)
!!
!
!
!
Line B
Line A
Math 100 !
Chapter 5 Review WKD
!
© 2016 Eitel
Solve each system of equations. List your answers as an ordered par.
7.
Equation A ⎧⎪ −4x + 2y = 20
!
⎨
Equation B ⎪ 3x + y = −5
⎩
Multiply Equation B by − 2 so that
Equation A has 2y and
Equation B has − 2y
⎧−4x + 2y = 20
⎨
−2 ⎩ 3x + y = −5
− 4x + 2y = 20
− 6x − 2y = 10
− 10x = 30
x= −3
Plug x = −3 into either equation A
and solve for y
Equation A
−4(x) + 2y = 20
−4(−3) + 2y = 20
12 + 2y = 20
2y = 8
y=4
Answer: (− 3, 4 )
Math 100 !
Chapter 5 Review WKD
!
© 2016 Eitel
8.
Equation A ⎧ x − 2y = −5
⎨
Equation B ⎩2x − 4y = 6
Multiply Equation A by − 2 so that
Equation A has –2x and
Equation B has − 2x
−2 ⎧ x − 2y = −5
⎨
⎩2x − 4y = 6
− 2x + 4y = 10
2x − 4y = 6
0 = 16
STOP: Both the x and y terms canceled out
and the remaining equation 0 = −2 is false
The lines are parallel,they have no common points.
Answer: No Solution
Math 100 !
Chapter 5 Review WKD
!
© 2016 Eitel
11.
Equation A ⎧⎪ 2x + 4y = −6
!
⎨
Equation B ⎪ x = 2y − 5
⎩
Equation B has been solved for x.
Substitute the expression x is equal to (2y − 5)
into Equation A in place of y.
Equation A with 2y − 5 subtituted for x
2(2y − 5) + 4y = −6
Solve for y
4y − 10 + 4y = −6
8y − 10 = −6
8y = 4
y=
1
2
!
1
into Equation B to find x
2
x = 2(y) − 5
plug y =
⎛ 1⎞
x = 2⎜ ⎟ − 5
⎝ 2⎠
x = 1− 5
x = −4
1⎞
⎛
Answer: ⎜ −4, ⎟
⎝
2⎠
Math 100 !
Chapter 5 Review WKD
!
© 2016 Eitel
12.
Equation A ⎧⎪ 8x − 4y = 20
!
⎨
Equation B ⎪ y = 2x − 5
⎩
Equation B has been solved for y.
Substitute the expression x is equal to (2x − 5)
into Equation A in place of y.
Equation A with 2x − 5 subtituted for y
8x − 4(2x − 5) = 20
Solve for y
8x − 8x + 20 = 20
20 = 20
Stop: The x term canceled out and
the remaining eqution 20 = 20 is true
Both equations describe the same line
any point on 8x − 4y = 20
would also be on y = 2x − 5
Answer: All Points on 8x − 4y = 20
or
Answer: All Points on y = 2x − 5
either one of the above is correct
Math 100 !
Chapter 5 Review WKD
!
© 2016 Eitel
14.
Equation A ⎧⎪ 2x − 4y = 10
⎨
Equation B ⎪ −x + 2y = −5
⎩
Multiply Equation B by 2
to eliminate the x terms
⎧⎪ 2x − 4y = 10
⎨
2 ⎪ −x + 2y = −5
⎩
2x − 4y = 10
−2x + 4y = −10
0=0
STOP: Both the x and y terms canceled out
and the remaining equation 0 = 0 is true
Both equations describe the same line
any point on 2x − 4y = 10
would also be on − x + 2y = −5
Answer: All Points on 2x − 4y = 10
or
Answer: All Points on − x + 2y = −5
either one of the above is correct
Math 100 !
Chapter 5 Review WKD
!
© 2016 Eitel
15.
Equation A ⎧⎪ 4x + 2y = 5
⎨
Equation B ⎪ 2x + y = −1
⎩
Multiply Equation B by − 2
to eliminate the x terms
⎧ 4x + 2y = 5
⎨
−2 ⎩2x + y = −1
4x + 2y = 5
−4x − 2y = −2
0=3
!
STOP: Both the x and y terms canceled out
and the remaining equation 0 = 3 is false
The lines are parallel , they have no common points.
Answer: No Solution
Math 100 !
Chapter 5 Review WKD
!
© 2016 Eitel
Equation A ⎧⎪ 3x − 5y = 11
⎨
Equation B ⎪ 2x − 6y = 2
⎩
17.
You must mutiply both rows by
different numbers to eliminate a variable
Multiply Equation A by − 2
Multiply Equation B by 3
to eliminate the x terms
−2 ⎧ 3x − 5y = 11
⎨
3 ⎩2x − 6y = 2
−6x + 10y = −22
6x − 18y = 6
− 8y = −16
y=2
Now add the two equations
Solve for y
Plug y = 2 into either equation A or B
and solve for y
Equation A
3(x) − 5(y) = 11
3(x) − 5(2) = 11
3x − 10 = 11
3x = 21
x=7
Answer:
( 7,2 )
Math 100 !
Chapter 5 Review WKD
!
© 2016 Eitel
18.
⎧ x y
+ = −1
Equation A ⎪ 2 6
⎪⎪
⎨
Equation B ⎪ −x y −1
⎪
+ =
⎪⎩ 2 3 2
Multiply Equation A by 6
Multiply Equation B by 6
to eliminate the fractions
6 ⎧ x + y = −1
⎪⎪ 2 6 1
⎨
−x y −1
6 ⎪⎪ 2 + 3 = 2
⎩
to get 2 new equations
Equation C ⎧ 3x + y = −6
⎨
Equation D ⎩−3x + 2y = −3
which you now can solve
Add Equation C and Equation D to eliminate
the x terms and solve for y
Equation C ⎧ 3x + y = −6
⎨
Equation D ⎩−3x + 2y = −3
3y = −9
y = −3
Plug y = −3 into either equation C or D
and solve for x
Equation C
3(x) + (y) = −6
3x − 3 = −6
3x = −3
x = −1
Answer: ( −1,−3)
Math 100 !
Chapter 5 Review WKD
!
© 2016 Eitel
21. Tom runs and walks for a total of 24 miles. The number of miles he runs is three times the
number of miles he walks. How far does he run and walk. Show all work
Sentence
Sentence
Sentence
Tom runs and walks
for a total of 24 miles
If we ADD the number of miles run(R)
and walks (W) the sum is 24
R + W = 24
The number of miles he runs is
three times
the number of miles he walks
The number of miles he runs (R) is =
R=
three times the number of miles he walks
3W
R = 3W
⎧ R + W = 24
⎨
⎩ R = 3W
substitutue the 3W for R in the top equation
3W + (W ) = 24
3W + W = 24
substitutue 3D for W
solve for W
3W + W = 24
4W = 24
W =6
plug W = 4 into the second equation and solve for D
R = 3W and W = 6
R = 3(6)
R = 18
ANSWER : He runs 18 miles and walks 6 miles
Check
18 + 6 = 24
18 = 3(6)
Math 100 !
Chapter 5 Review WKD
!
© 2016 Eitel