Graph each system and determine the number of solutions that it has. If it has one solution, name it.
1.y = 2x
y = 6 − x
SOLUTION:
To graph the system, write both equations in slope-intercept form.
y = 2x
y = −x + 6
The graph appears to intersect at the point (2, 4). You can check this by substituting 2 for x and 4 for y.
The solution is (2, 4).
2.y = x − 3
y = −2x + 9
SOLUTION:
y = x − 3
y = −2x + 9
The graph appears to intersect at the point (4, 1). You can check this by substituting 4 for x and 1 for y.
The solution is (4, 1).
3.x − y = 4
x + y = 10
SOLUTION:
To graph the system, write both equations in slope-intercept form.
Equation 1:
Equation 2:
Graphandsolve.
y = x − 4
y = −x + 10
The graph appears to intersect at the point (7, 3). You can check this by substituting 7 for x and 3 for y.
The solution is (7, 3).
4.2x + 3y = 4
2x + 3y = −1
SOLUTION:
To graph the system, write both equations in slope-intercept form.
Equation 1:
Equation 2:
Graph and solve.
The lines are parallel. So, there is no solution.
Use substitution to solve each system of equations.
5.y = x + 8
2x + y = −10
SOLUTION:
y = x + 8
2x + y = −10
Substitute x + 8 for y in the second equation.
Use the solution for x and either equation to find the value for y.
The solution is (−6, 2).
6.x = −4y − 3
3x − 2y = 5
SOLUTION:
x = −4y − 3
3x − 2y = 5
Substitute −4y − 3 for x in the second equation.
Use the solution for y and either equation to find the value for x.
The solution is (1, −1).
7.GARDENING Corey has 42 feet of fencing around his garden. The garden is rectangular in shape, and its length is
equal to twice the width minus 3 feet. Define the variables, and write a system of equations to find the length and
width of the garden. Solve the system by using substitution.
SOLUTION:
Sample answer: Let w be the width and let bethelength.Then,2w + 2 = 42 and =2w − 3. Substitute 2w − 3
for inthefirstequation.
Use the solution for w and either equation to find the value for .
The width of the garden is 8 feet and the length is 13 feet.
8.MULTIPLECHOICE Use elimination to solve the system.
6x − 4y = 6
−6x + 3y = 0
A(5, 6)
B(−3, −6)
C(1, 0)
D(4, −8)
SOLUTION:
Because 6x and −6x have opposite coefficients, add the equations.
Now,substitute−6 for y in either equation to find the value of x.
The solution is (−3, −6). So, the correct choice is B.
9.SHOPPING Shelly has $175 to shop for jeans and sweaters. Each pair of jeans costs $25, each sweater costs $20,
and she buys 8 items. Determine the number of pairs of jeans and sweaters Shelly bought.
SOLUTION:
Let j = the number of pairs of jeans and s = the number of sweaters. Then, j + s = 8 and 25j + 20s = 175.
Solve the first equation for j.
Substitute 8 – s for j in the second equation.
Now, substitute 5 for s in either equation to find the value of j.
Shelly bought 3 pairs of jeans and 5 sweaters.
Use elimination to solve each system of equations.
10.x + y = 13
x − y = 5
SOLUTION:
Because y and −y have opposite coefficients, add the equations.
Now, substitute 9 for x in either equation to find the value of y.
The solution is (9, 4).
11.3x + 7y = 2
3x − 4y = 13
SOLUTION:
Because 3x and 3x have the same coefficients, multiply equation 2 by −1, then add the equations.
Add the equations.
Now, substitute −1 for y in either equation to find the value of x.
The solution is (3, −1).
12.x + y = 8
x − 3y = −4
SOLUTION:
Because x and x have the same coefficients, multiply equation 2 by –1 and then add the equations.
Add the equations.
Now, substitute 3 for y in either equation to find the value of x.
The solution is (5, 3).
13.2x + 6y = 18
3x + 2y = 13
SOLUTION:
Multiply the second equation by −3.
Now, because 6y and −6y have opposite coefficients, add the equations.
Now, substitute 3 for x in either equation to find the value of y.
The solution is (3, 2).
14.MAGAZINES Julie subscribes to a sports magazine and a fashion magazine. She received 24 issues this year. The
number of fashion issues is 6 less than twice the number of sports issues. Define the variables, and write a system of
equations to find the number of issues of each magazine.
SOLUTION:
Let f = the number of fashion issues and s = the number of sports issues. So, f + s = 24 and f = 2s – 6.
Substitute 2s – 6 for f in the first equation.
Now, substitute 10 for s in either equation to find the value of f.
So, Julie received 14 fashion issues and 10 sports issues.
Determine the best method to solve each system of equations. Then solve the system.
15.y = 3x
x + 2y = 21
SOLUTION:
y = 3x
x + 2y = 21
Substitute 3x for y in the second equation.
Use the solution for x and either equation to find the value for y.
The solution is (3, 9).
16.x + y = 12
y = x − 4
SOLUTION:
y = x – 4
x + y = 12
Substitute x – 4 for y in the second equation.
Use the solution for x and either equation to find the value for y.
The solution is (8, 4).
17.x + y = 15
x − y = 9
SOLUTION:
Because the y-terms have opposite coefficients, add the equations.
Now,substitute12 for x in either equation to find the value of y.
The solution is (12, 3).
18.3x + 5y = 7
2x − 3y = 11
SOLUTION:
Because none of the terms are opposites, use elimination by multiplication to solve. Multiply the first equation by 2
and the second equation by -3. Then add the equations to eliminate the x-term.
3x + 5y = 7
2x − 3y = 11
Substitute -1 for y in the second equation to find x.
The solution is (4, –1).
19.OFFICE SUPPLIES At a sale, Ricardo bought 24 reams of paper and 4 inkjet cartridges for $320. Britney bought
2 reams of paper and
1 inkjet cartridge for $50. The reams of paper were all the same price and the inkjet cartridges were all the same
price. Write a system of
equations to represent this situation. Determine the best method to solve the system of equations. Then solve the
system.
SOLUTION:
24p + 4c = 320
2p + c = 50
Solve equation 2 for c.
c = 50 – 2p
Substitute 50 – 2p for c in the other equation.
Substitute 7.5 for p in equation 2.
c = 50 – 2(7.5)
c = 50 – 15
c = 35
paper: $7.50; cartridge: $35
Solve each system of inequalities by graphing.
20.x > 2
y < 4
SOLUTION:
Grapheachinequality.
The graph of x > 2 is dashed and is not included in the graph of the solution.
Thegraphofy<4isalsodashedandisnotincludedinthegraphofthesolution.
The solution of the system is the set of ordered pairs in the intersection of the graphs of x > 2 and y < 4. Overlay the
graphs and locate the green region. This is the intersection.
The solution region is shaded in gray.
21.x + y≤5
y ≥x + 2
SOLUTION:
Grapheachinequality.
The graph of x + y≤5issolidandisincludedinthegraphofthesolution.
The graph of y≥x + 2 is also solid and is included in the graph of the solution.
The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y≤5andy≥x + 2.
Overlay the graphs and locate the green region. This is the intersection.
The solution region is shaded in gray.
22.3x − y > 9
y > −2x
SOLUTION:
Grapheachinequality.
The graph of 3x − y>9isdashedandisnotincludedinthegraphofthesolution.
The graph of y > −2xisalsodashedandisnotincludedinthegraphofthesolution.
The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x − y > 9 and y > −2x.
Overlay the graphs and locate the green region. This is the intersection.
The solution region is shaded in gray.
23.y ≥2x + 3
−4x − 3y > 12
SOLUTION:
Graph each inequality.
Thegraphofy≥2x+3issolidandisincludedinthegraphofthesolution.
The graph of −4x − 3y>12isdashedandisnotincludedinthegraphofthesolution.
The solution of the system is the set of ordered pairs in the intersection of the graphs of y≥2x + 3 and −4x − 3y >
12. Overlay the graphs and locate the green region. This is the intersection.
The solution region is shaded in gray.
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Practice Test - Chapter 6
