Download Test on Topic 15 Solving a System of Equations

Test on Topic 15 System of Equations
Test on Topic 15 System of Equations
Determine which of the following points, A(2,1) , B(1, – 3) and C(– 1 , 3) if any, satisfy both
pairs of equations.
1.
4x – 7
4
y =
3x – 2y =
2.
Determine which of the following points, A(1,1) , B(– 1, – 3) and C(0, 1) if any, satisfy both pairs of
equations.
y = 2x – 1
2x – 4y = – 2
3(a).
On the grid opposite graph 2x + 4y = 0 by plotting 3 points.
x
0
2
4
3(b).
y
y
On the grid opposite graph y = 2x – 5
by plotting 3 points.
x
0
x
y
0
1
2
3.(c)
4.
Solution to the system of equations is?
Identify each system of linear equations below as consistent, inconsistent or dependant.
Also indicate the number of solutions that would occur.
Page | 1
Test on Topic 15 System of Equations
5.
Express each of the system of equations below in the slope intercept form ( y = ….. form).
And without drawing graphs identify each system of linear equations below as consistent,
inconsistent or dependant. Also indicate the number of solutions that would occur.
(a) 2y – 3x =
4x
=
6.
7.
8.
9.
(b) 4y – 8x =
2y – 6 =
4
8y – 7
12
4x
Express each of the system of equations below in the slope intercept form ( y = ….. form).
And without drawing graphs identify each system of linear equations below as consistent,
inconsistent or dependant. Also indicate the number of solutions that would occur.
(a) y – 2x = 4
2y = 4 – 2x
(b) x – 2y =
2x =
Solve the system of equations
y =
2x + 3y =
Solve the system of equations
Solve the system of equations
10. Solve the system of equations
y
y
x – 2y
3x + y
12. Solve the system of equations
13. Solve the system of equations
14. Solve the system of equations
3x – 3
13
(c) x – 4
4y – 8
=
=
8
8
=
=
by using the substitution method.
2x + 3
4x – 8
=
=
–8
y – 2x
by using the addition method.
by using the substitution method.
6
4
by using the addition method.
5x – 2y =
5 =
–7
y – 3x
by using the substitution method.
2x – 2y =
4x + 3y =
4
1
by using the addition method.
5x – 2y =
3x + 2y =
–8
24
4y
2x
by using the substitution method.
5x – 2y =
2x + 4y =
2x – 2y =
3 =
11. Solve the system of equations
10
4y + 20
by using the addition method.
15. Solve the system of equations
2x – 2y =
4x + 3y =
4
1
by using the addition method.
16. Solve the system of equations
5x – 2y =
3x + 2y =
10
14
by using the addition method.
17. Solve the system of equations
2x – 3y =
3x + 4y =
–3
21
by using the addition method.
18. Solve the system of equations
2x – 2y =
4x + 3y =
4
1
by using the addition method.
Page | 2
Test on Topic 15 System of Equations
19. In 1997, France was the most visited country in the world and U.S.A. was the second most visited.
A total of 116 million people visited these countries. If 18 million more visited France than the
U.S.A. how many people visited each country.
20. The plumber charges a fixed rate for turning up at your house plus a charge per
hour for the work done From previous jobs that the plumber has done it was found that a 4 hour job
will cost a total of $283 while a 6 hour job will cost a total of $387.
What is the fixed rate and the charge per hour?
21. The Acme printing company charges a fixed setup fee x, plus a fixed amount per page y.
John has used this company before and after looking up his receipts he finds that his last two printing
jobs cost the following. A 2000 page job cost him $150, while a 3000 page job cost him $200.
(a) Set up a system of equations to reflect the above information.
(b) Solve the system of equations and so find how much the fixed setup fee x is and how much is the
charge per page y?
(c) He needs to have 4500 copies of a flyer made, what should the Acme Printing company charge
him to do this job.
22. A nutritionist finds that a large order of fries has 20 grams more than twice the fat content of a
Haggis Burger. The difference in fat content between the fries and a Haggis Burger is 100 grams.
What is the fat content of a large order of fries and a Haggis Burger?
23. By weight one alloy of brass is 70% Copper and 30% Zinc. Another Alloy is 40% Copper and 60%
Zinc. How many grams of each alloy would need to be melted and combined to obtain 600 grams of
a brass alloy that is 60% Copper and 40% Zinc?
24. A truck rental agency charges a daily fee plus a mileage fee. Julie was charged $85 for two days and
100 miles and Christina was charged $165 for 3 days and 400 miles. What 8is the agency’s daily fee
and what is the mileage fee?
25. A pet store owner wants to make 600 pounds of mixed bird seed, The sunflower seeds cost $2.30 per
pound and the peanut seeds cost $1.40 per pound. The total cost of all the seed used in the mixture
was $1200. How much of each seed was used?
26. The fat content of a haggis is more than that for fries.
The total fat content for a haggis and fries is 190 grams. The amount of fat in a haggis is 40 grams
more than twice the amount of fat that is in fries. Find the amount of fat that is in a haggis and in a
portion of fries?
Page | 3
Test on Topic 15 System of Equations
Test on Topic 15 System of Equations Solutions
1.
Determine which of the following points, A(2,1) , B(1, – 3) and C(– 1 , 3) if any, satisfy both
pairs of equations.
y =
3x – 2y =
For A(2,1)
For C(– 1 , 3)
2.
=
=
=
4x – 7
4(2) – 7
1
(yes)
For A(2,1)
y =
–3 =
–3 =
4x – 7
4(1) – 7
–3
(yes)
For B(1, – 3)
y
3
3
4x – 7
4(– 1) – 7
– 11
(no)
y
1
1
For B(1, – 3)
4x – 7
4
=
=
=
3x – 2y =
3(2) – 2(1) =
4 =
4
4
4
(yes)
3x – 2y =
3(1) – 2(– 3) =
9 =
4
4
4
(no)
For C(– 1 , 3)
3x – 2y =
3(– 1) – 2(3) =
–9
=
4
4
4
(no)
Determine which of the following points, A(1,1) , B(– 1, – 3) and C(0, 1) if any, satisfy both pairs of
equations.
y = 2x – 1
2x – 4y = – 2
For A(1,1)
y
=
=
= 2x – 1
2(1) – 1
1 true
2x – 4y =
2(1) – 4(1) =
–2 =
–2
–2
– 2 true
y
–3 =
–3 =
= 2x – 1
2(–1) – 1
– 3 true
2x – 4y =
2(–1) – 4(–3) =
10 =
–2
–2
– 2 false
x = 1 and y = 1
1
1
For B(– 1, – 3)
For C(0,1)
x = –1 and y = –3
x = 0 and y = 1
y
1
1
=
=
=
2x – 1
2(0) – 1
–1 false
So the point A(1,1) satisfies the pair of equations.
Page | 4
Test on Topic 15 System of Equations
3(a).
On the grid opposite graph 2x + 4y = 0 by plotting 3 points.
x
0
2
4
3(b).
3.(c)
4.
On the grid opposite graph y = 2x – 5
by plotting 3 points.
x
y
0
–5
1
–3
2
–1
0
x
2x + 4y = 0
Solution is (2, – 1)
Identify each system of linear equations below as consistent, inconsistent or dependant.
Also indicate the number of solutions that would occur.
Consistent
# of solutions = 1
5.
y = 2x – 5
y
y
0
–1
–2
Inconsistent
# of solutions = o
dependant
# of solutions = infinite
Express each of the system of equations below in the slope intercept form ( y = ….. form).
And without drawing graphs identify each system of linear equations below as consistent,
inconsistent or dependant. Also indicate the number of solutions that would occur.
(a) 2y – 3x =
4x
=
2y – 3x =
2y =
y
=
Answer :
4
8y – 7
4
3x + 4
3
x2
2
4x =
4x + 7 =
1
7
x =
2
2
8y – 7
8y
y
These two lines have different slopes, so the equations are consistent and there
will be one solution
Page | 5
Test on Topic 15 System of Equations
(b) 4y – 8x =
2y – 6 =
12
4x
4y – 8x =
4y =
y =
Answer :
6.
2y – 6 =
2y =
y =
12
8x + 12
2x  3
These two lines are identical, so the equations are dependant and there
will be infinite solutions.
Express each of the system of equations below in the slope intercept form ( y = ….. form).
And without drawing graphs identify each system of linear equations below as consistent,
inconsistent or dependant. Also indicate the number of solutions that would occur.
(a) y – 2x = 4
2y = 4 – 2x
(b) x – 2y =
2x =
10
4y + 20
x – 2y
– 2y
y
10
– x + 10
½x – 5
y – 2x = 4
y = 2x + 4
2y =
y =
4 – 2x
2–x
consistent 1 solution
7.
4x
4x + 6
2x + 3
Solve the system of equations
Solution:
Substitute y = 3x – 3 into the equation
Substitute x = 2 into equation
=
=
=
(c) x – 4
4y – 8
2x = 4y + 20
2x – 20 = 4y
½x–5 = y
dependant infinite solution
=
=
4y
2x
x–4
=
¼x–1=
4y
y
4y – 8
4y
y
2x
2x + 8
½x + 2
=
=
=
inconsistent 0 solutions
y = 3x – 3 by using the substitution method.
2x + 3y = 13
2x + 3y
2x + 3(3x – 3)
2x + 9x – 9
11x – 9
11x
x
=
=
=
=
=
=
13
13
13
13
22
2
y = 3x – 3 = 3(2) – 3 = 3
So the solution is x = 2 and y = 3 or (2,3)
Page | 6
Test on Topic 15 System of Equations
8.
5x – 2y =
2x + 4y =
Solve the system of equations
Solution:
5x – 2y =
2x + 4y =
8
8
equation1….. multiply by 2
equation 2
Substitute x = 2 into equation
2x + 4y
2(2) + 4y
4 + 4y
4y
y
=
=
=
=
=
8
8
by using the addition method.
10x – 4y =
2x + 4y =
12x
=
x =
16
8
24
2
add equationd
8
8
8
4
1
So the solution is x = 2 and y = 1 or (2,1)
9.
Solve the system of equations
y
Put y = 2x + 3 into the equation
Put x = 5.5 into the equation
y
2x + 3
3
11
5.5
y
y
y
y
= 2x + 3
4x – 8
by using the substitution method.
=
=
=
=
=
4x – 8
4x – 8
2x – 8
2x
x
=
=
=
=
4x – 8
4(5.5) – 8
22 – 8
14
2x – 2y = – 8
3 = y – 2x
10. Solve the system of equations
Rearrange the equation 3 =
y
=
Solution is (5.5,14)
by using the substitution method.
y – 2x so that it is in the form y = …..
3 = y – 2x
3 + 2x = y
Put y = 3 + 2x into the equation
2x – 2y
2x – 2(3 + 2x)
2x – 6 – 4x
– 2x – 6
– 2x
x
Put x = 1 into equation y = 3 + 2x = 3 + 2(1) = 5
=
=
=
=
=
=
–8
–8
–8
–8
–2
1
Solution is (1,5)
Page | 7
Test on Topic 15 System of Equations
11. Solve the system of equations
x – 2y
3x + y
=
=
6
4 multiply by 2
Put x = 2 into
3x + y
3(2) + y =
6+y
y =
Solution is (2, – 2 )
x – 2y
3x + y =
=
4
x – 2y
6x + 2y
7x
x
6
8
14
2
=
=
=
=
6
by using the addition method.
= 4
4
= 4
–2
12. Solve the system of equations
Re arrange 5 =
y – 3x
Substitute y =
3x + 5 into
to get
5x – 2y =
5
–7
by using the substitution method.
= y – 3x
5
y – 3x
y
y – 3x
5
3x + 5
=
=
=
5x – 2y
5x – 2(3x + 5)
5x – 6x – 10
– x – 10
–x
x
=
=
=
=
=
=
–7
–7
–7
–7
3
–3
Use x = – 3 in equation y = 3x + 5 = 3(– 3) + 5 = – 9 + 5 = – 4
The solution to the above system of equations is (– 3, – 4)
2x – 2y =
4x + 3y =
13. Solve the system of equations
Equation 1:
Equation 2 :`
2x – 2y =
4x + 3y =
Use x = 1 in the equation
4
1
4
1
by using the addition method.
6x – 6y
8x + 6y
14x
x
multiply by 3
multiply by 2
4x + 3y
4(1) + 3y
4 + 3y
3y
y
=
=
=
=
=
=
=
=
=
12
2
14
1
(add)
1
1
1
–3
–1
The solution to the above system of equations is (1, – 1)
Page | 8
Test on Topic 15 System of Equations
5x – 2y =
3x + 2y =
14. Solve the system of equations
5x – 2y
3x + 2y
8x
x
=
=
=
=
–8
24
16
2
–8
24
by using the addition method.
(add)
Use x = 2 in the equation
3x + 2y
3(2) + 2y
6 + 2y
2y
y
=
=
=
=
=
24
24
24
16
8
The solution to the above system of equations is (2, 8)
2x – 2y =
4x + 3y =
15. Solve the system of equations
Equation 1:
Equation 2 :`
2x – 2y =
4x + 3y =
Use x = 1 in the equation
4
1
3x + 5 into
by using the addition method.
6x – 6y
8x + 6y
14x
x
multiply by 3
multiply by 2
4x + 3y
4(1) + 3y
4 + 3y
3y
y
The solution to the above system of equations is
Re arrange 5 = y – 3x to get
5 =
y – 3x =
y
=
Substitute y =
4
1
=
=
=
=
=
=
=
=
=
12
2
14
1
(add)
1
1
1
–3
–1
(1, – 1)
y – 3x
5
3x + 5
5x – 2y
5x – 2(3x + 5)
5x – 6x – 10
– x – 10
–x
x
=
=
=
=
=
=
–7
–7
–7
–7
3
–3
Use x = – 3 in equation y = 3x + 5 = 3(– 3) + 5 = – 9 + 5 = – 4
The solution to the above system of equations is (– 3, – 4)
Page | 9
Test on Topic 15 System of Equations
16. Solve the system of equations
5x – 2y =
3x + 2y =
10
14
8x
x
=
=
=
=
=
=
=
14
14
14
5
2.5
2x – 3y =
3x + 4y =
–3
21
Add the equations
Put x = 3 into equation
3x + 2y
3(3) + 2y
9 + 2y
2y
y
by using the addition method.
24
3
So the Solution is (3,2.5)
17. Solve the system of equations
2x – 3y =
3x + 4y =
– 3 (multiply by 4)
21 (multiply by 3)
Add the equations:
Put x = 3 into the equation
3x + 4y
3(3) + 4y
9 + 4y
4y
y
8x – 12y
9x + 12y
17x
x
=
=
=
=
=
by using the addition method.
=
=
=
=
– 12
63
51
3
21
21
21
12
3
Answer: Solution to system of equations is (3,3)
2x – 2y =
4x + 3y =
18. Solve the system of equations
Equation 1:
Equation 2 :`
2x – 2y =
4x + 3y =
Use x = 1 in the equation
4
1
4
1
by using the addition method.
6x – 6y
8x + 6y
14x
x
multiply by 3
multiply by 2
4x + 3y
4(1) + 3y
4 + 3y
3y
y
=
=
=
=
=
=
=
=
=
12
2
14
1
(add)
1
1
1
–3
–1
The solution to the above system of equations is (1, – 1)
Page | 10
Test on Topic 15 System of Equations
19. In 1997, France was the most visited country in the world and U.S.A. was the second most visited.
A total of 116 million people visited these countries. If 18 million more visited France than the
U.S.A. how many people visited each country.
Let x
y
=
=
x+y
x–y
2x
x
the number of visitors to France (in millions)
the number of visitors to U.S.A. (in millions)
=
=
=
=
116
18
134
67
(add)
Use x = 67 in equation
x+y
67 + y
y
=
=
=
116
116
49
So France has 67 million visitors and U.S.A. has 49 million visitors
20. The plumber charges a fixed rate for turning up at your house plus a charge per
hour for the work done From previous jobs that the plumber has done it was found that a 4 hour job
will cost a total of $283 while a 6 hour job will cost a total of $387.
What is the fixed rate and the charge per hour?
Let x
y
=
=
fixed amount
charge per hour
x + 4y
x + 6y
=
=
283
387
Rearrange
x + 4y
=
283
Use x = 283 – 4y into equation
x = 283 – 4y
to get
x + 6y
283 – 4y + 6y
283 + 2y
2y
y
=
=
=
=
=
387
387
387
104
52
Use y = 52 in equation x = 283 – 4y = 283 – 4(52) = 283 – 208 = 75
So the plumber had a fixed amount of $75 plus he charged $52 per hour.
Page | 11
Test on Topic 15 System of Equations
21. The Acme printing company charges a fixed setup fee x, plus a fixed amount per page y.
John has used this company before and after looking up his receipts he finds that his last two printing
jobs cost the following. A 2000 page job cost him $150, while a 3000 page job cost him $200.
We set up a system of equations to reflect the above information.
(b) 2000 page job cost him $150 gives us the equation
3000 page job cost him $200 gives us the equation
x + 2000y
x + 3000y
=
=
150
200
(c) Solve the system of equations and so find how much the fixed setup fee x is and how much is the
charge per page y?
x + 2000y = 150
can be written as
x = – 2000y + 150
Substitute x = – 2000y + 150 into the equation
Substitute y =
0.5 into the equation x
x
x
x + 3000y
– 2000y + 150 + 3000y
150 + 1000y
100y
y
=
=
=
=
=
200
200
200
50
0.5
= – 2000y + 150
= – 2000(0.5) + 150
= 50
So set up f = x = $50 and cost pr page = y = $0.50
(d) He needs to have 4500 copies of a flyer made, what should the Acme Printing company charge
him to do this job.
Cost
=
=
=
x + 4500y
5 0 + 4500(0.5)
$275
Page | 12
Test on Topic 15 System of Equations
22. A nutritionist finds that a large order of fries has 20 grams more than twice the fat content of a
Haggis Burger. The difference in fat content between the fries and a Haggis Burger is 100 grams.
What is the fat content of a large order of fries and a Haggis Burger?
Let x
y
=
=
number of grams of fat in an order of fries
number of grams of fat in an Haggis Burger
The information that A large order of fries has 20 grams more than twice the fat content of a Haggis
Burger, can be translated into x = 2y + 20
The information The difference in fat content between the fries and a Haggis Burger is 100 grams.
Can be translated into x – y = 100
( x is larger than y so this is the correct order)
Put these equations together and solve.
x = 2y + 20
x – y = 100
Put y = 2x + 20 into the equation
x–y
2y + 20 – y
y + 20
y
=
=
=
=
100
100
100
80
Put y = 80 into x = 2y + 20 = 2(80) + 20 = 180
Answer:
The solution to this problem is
and
x = number of grams of fat in an order of fries = 180 grams
y = number of grams of fat in an Haggis Burger = 80 grams
Page | 13
Test on Topic 15 System of Equations
23. By weight one alloy of brass is 70% Copper and 30% Zinc. Another Alloy is 40% Copper and 60%
Zinc. How many grams of each alloy would need to be melted and combined to obtain 600 grams of
a brass alloy that is 60% Copper and 40% Zinc?
Let x = amount of alloy 1 used
Let y = amount of alloy 2 used
The information that alloys combined to obtain 600 grams of a brass alloy gives us the equation
x + y = 600
The information that alloys combined to give us 60% copper gives us the equation
0.7x + 0.4y = 0.6(600) which in turn becomes 0.7x + 0.4y = 360
Put these equations together and solve.
x+y
=
0.7x + 0.4y =
Rearrange
x+y
y
Put y = – x + 600in to the equation
0.7x + 0.4y
0.7x + 0.4(– x + 600)
0.7x – 0.4x + 240
0.3x + 240
0.3x
x
600
360
=
=
600
– x + 600
=
=
=
=
=
=
360
360
360
360
120
400
Put x = 400 in to y = – x + 600 = – 400 + 600 = 200
Answer:
The solution to this problem is x = amount of alloy 1 used = 400 grams
and
y = amount of alloy 2 used = 200 grams
24. A truck rental agency charges a daily fee plus a mileage fee. Julie was charged $85 for two days and
100 miles and Christina was charged $165 for 3 days and 400 miles. What 8is the agency’s daily fee
and what is the mileage fee?
Julie was charged $85 for two days and 100 miles
Christina was charged $165 for 3 days and 400 miles
– 6d – 300m
6d + 800m
500m
m
Put m = 0.15 into the equation
2d + 100m = 85
2d + 15 = 85
2d = 70
d = 35
So solution is d = $35 per day and m = $0.15 per mile
2d + 100m
3d + 400m
=
=
85
165
multiply by – 3
multiply by 2
Add the equations
2d + 100m
3d + 400m
=
=
=
=
=
=
85
165
– 255
330
75
0.15
Page | 14
Test on Topic 15 System of Equations
25. A pet store owner wants to make 600 pounds of mixed bird seed, The sunflower seeds cost $2.30 per
pound and the peanut seeds cost $1.40 per pound. The total cost of all the seed used in the mixture
was $1200. How much of each seed was used?
x = amount of sunflower seed used
and
y = amount of peanut seed used
Equation two is formed by using the fact that he “wants to make 600 pounds of mixed bird seed”
This corresponds to the equation x + y = 600
Equation one is formed by the fact that “total cost of all the seed used in the mixture was $1200”
This corresponds to the equation 2.3x + 1.4y = 1200
x+y
=
2.3x + 1.4y =
2.3x + 1.4y
2.3x + 1.4(– x + 600)
2.3x – 1.4x + 840
0.9x + 840
0.9 x
x
=
=
=
=
=
=
Also written as y = – x + 600
600
1200
1200
1200
1200
1200
360
400
Substitute x = 400 into the equation y = – x + 600 = – 400 + 600 = 200
So amount of sunflower seed used = 400 pounds and amount of peanut seed used is 200 pounds
26. The fat content of a haggis is more than that for fries.
The total fat content for a haggis and fries is 190 grams. The amount of fat in a haggis is 40 grams
more than twice the amount of fat that is in fries. Find the amount of fat that is in a haggis and in a
portion of fries?
Let x
=
fat content of a haggis
y
=
fat content of fries
The total fat content for a haggis and fries is 190 grams gives us the equation
x+y
=
190
The amount of fat in a haggis is 40 grams more than twice the amount of fat that is in fries gives us
the equation.
x = 2y + 40
Solve by the substitution method: put x = 2y + 40 into equation
x+y
2y + 40 + y
3y + 40
3y
y
=
=
=
=
=
190
190
190
150
50
Put y = 50 into equation
x+y
= 190
x+ 50 = 190
x = 140
The Solution is the fat content of a haggis = 140 grams and the fat content of the fries = 50 grams
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