Question 1

Solve by the substitution method.
-x - 4y = -4 7x - 3y = -28
A)(-4, 0)
B)(-3, -4)
C)(4, -1)
D)No solution

Accepted Answer

Solving the first equation for x gives x = 4y - 4. Substituting this into the second equation gives 7(4y - 4) - 3y = -28, which simplifies to 28y - 28 - 3y = -28, or 25y = 0, so y = 0. Substituting y = 0 into x = 4y - 4 gives x = -4. Therefore, the solution is (-4, 0).

Question 2

Solve using the elimination method.
-- x - 6y = - 11 3x + 6y = 9
A)(-1, 2)
B)(-2, -1)
C)(0, 1)
D)No solution

Accepted Answer

Adding the two equations directly eliminates y, leading to a solvable equation for x.

Question 3

Solve by the substitution method.
-x - 2y = -12 -2x - 3y = -4
A)(-4, 4)
B)(4, 5)
C)(-5, 5)
D)No solution

Accepted Answer

To solve the system of equations by substitution, we first solve one of the equations for one variable in terms of the other. Let's solve the first equation for x:x - 2y = -12x = 2y - 12Now, substitute this expression for x in the second equation:-2(2y - 12) - 3y = -4-4y + 24 - 3y = -4-7y = -28y = 4Now, substitute y = 4 back into one of the original equations to solve for x. Using the first equation:x - 2(4) = -12x - 8 = -12x = -4Therefore, the solution is (-4, 4).

Question 4

Solve by the substitution method.
-x + y = 6 8x + 8y = 48
A)(6, 8)
B)(0, 0)
C)Infinite number of solutions
D)(1, 5)

Accepted Answer

The second equation is a multiple of the first, indicating the same line, hence infinite solutions.

Question 5

Solve by the substitution method.
-x + y = -6 x + y = -4
A)No solution
B)(0, 0)
C)(-6, -4)
D)(0, -10)

Accepted Answer

The two equations are parallel lines (same slope, different intercepts), so they never intersect, meaning there is no solution.

Question 6

Solve by the substitution method.
-9x - 6y = -72 -2x + 3y = 31
A)No solution
B)(-3, 10)
C)(-2, 9)
D)(-2, 10)

Accepted Answer

Substitute x = -2 into the second equation: -2(-2) + 3y = 31, which simplifies to 4 + 3y = 31. Solving for y gives y = 9. Thus, the solution is (-2, 9).

Question 7

Solve by the substitution method.
-x - 6y = -6 4x - 5y = -5
A)No solution
B)(-1, 0)
C)(0, 1)
D)(1, 0)

Accepted Answer

The answer of Solve by the substitution method.
-x - 6y...

Question 8

Solve by the substitution method.
-7x + 7y = -28 -4x - 4y = 16
A)(-5, 1)
B)(-4, 0)
C)No solution
D)(-4, 1)

Accepted Answer

First, simplify both equations: x + y = -4 and -x - y = -4. These equations are equivalent, meaning they represent the same line. Solving x + y = -4 gives x = -4, y = 0.

Question 9

Solve the problem.
-The perimeter of a rectangle is 50 m. If the width were doubled and the length were increased by 5 m, the perimeter would be 74 m. What are the length and width of the rectangle?
A)width 7 m, length 12 m
B)width 12 m, length 12 m
C)width 7 m, length 18 m
D)width 18 m, length 7 m

Accepted Answer

Let's say the width is "w" and the length is "l". 
According to the problem, 
2l + 2w = 50 
l + w = 25 
Also, 
2(l+5) + 2(2w) = 74 (width doubled and length increased by 5) 
2l + 4w + 10 = 74 
2l + 4w = 64 
l + 2w = 32

Now we have two equations: 
l + w = 25 
l + 2w = 32

Solving for "l" and "w", we get: 
w = 7 and l = 18

Therefore, the width is 7 m and the length is 18 m. Let's represent the width with "w" and the length with "l".
From the problem, we have two equations:
2w + 2l = 50  --> equation 1 
2(w x 2) + (l + 5) x 2 = 74  --> equation 2

To solve for w and l, we need to solve for either variable in terms of the other. From equation 1, we can solve for l:
2w + 2l = 50
2l = 50 - 2w
l = 25 - w

Now we can substitute this into equation 2:
2(w x 2) + (l + 5) x 2 = 74
4w + 2(l + 5) = 74
4w + 2(25 - w + 5) = 74
2w + 60 = 74
2w = 14
w = 7

Now we can use equation 1 to solve for l:
2w + 2l = 50
2(7) + 2l = 50
14 + 2l = 50
2l = 36
l = 18

Therefore, the width is 7 m and the length is 18 m. So, the answer is C.

Question 10

Solve by the substitution method.
--7x + 7y = 21 -5x - 2y = -6
A)(0, 4)
B)No solution
C)(0, 3)
D)(-1, 4)

Accepted Answer

-7x+7y = 21 can be rewritten as x-y = -3.
Now, using substitution method:
-5x-2y = -6
-5x-2(3-x) = -6
-5x-6+2x = -6
-3x = 0
x = 0
Substituting this value of x in x-y=-3:
0-y=-3
y=3
So, the solution is (0,3), matching with choice C.

Question 11

Solve using the substitution method.
-x = -1 + 8y x + -8y = 3
A)Infinitely many solutions
B) 
C) 
D)No solution

Accepted Answer

The answer of Solve using the substitution method.
-x = -1...

Question 12

Solve the problem.
-The sum of two numbers is 34 and their difference is 8. Find the numbers.
A)15 and 23
B)21 and 13
C)27 and 7
D)19 and 15

Accepted Answer

Let's call the two numbers x and y. We know that x + y = 34 and x - y = 8. If we add these two equations, we get:
x + y + x - y = 34 + 8
2x = 42
x = 21
Now that we know x = 21, we can substitute this value into either equation to solve for y:
21 + y = 34
y = 13
Therefore, the two numbers are 21 and 13, which is answer choice B. Let's call the two numbers "x" and "y". We know that:
x + y = 34 (since the sum of the two numbers is 34)
x - y = 8 (since the difference of the two numbers is 8)
Now we can use these two equations to solve for x and y. One way to do this is to add the two equations together:
(x + y) + (x - y) = 34 + 8
2x = 42
x = 21
Now that we know x is 21, we can plug that into either equation to solve for y:
21 + y = 34
y = 13
So the two numbers are 21 and 13, which is answer choice B.

Question 13

Solve the problem.
-The sum of two angles is 110°. One angle is 25° less than twice the other. Find the angles.
A)43° and 61°
B)45° and 65°
C)61° and 49°
D)43° and 67°

Accepted Answer

Let's call one angle x and the other angle y. 
From the first sentence, we know that x + y = 110. 
From the second sentence, we know that x = 2y - 25. 
We can substitute x in terms of y into the first equation: 
(2y - 25) + y = 110 
Simplifying the equation, we get: 
3y - 25 = 110 
3y = 135 
y = 45 
We can substitute y = 45 into either equation to solve for x: 
x = 2(45) - 25 = 65 
Therefore, the angles are 45° and 65°, which is answer choice B.

Question 14

Solve the problem.
-The perimeter of a triangle is 48 cm. The triangle is isosceles now, but if its base were lengthened by 6 cm and each leg were shortened by 6 cm, it would be equilateral. Find the base of the original triangle.
A)8 cm
B)20 cm
C)14 cm
D)7 cm

Accepted Answer

The answer of Solve the problem.
-The perimeter of a triangle...

Question 15

Solve by the substitution method.
-7x + 27 = 8y 5x + 6y = -31
A)No solution
B)(-5, 0)
C)(-6, 0)
D)(-5, -1)

Accepted Answer

From the second equation, we can solve for y in terms of x:
5x + 6y = -31
6y = -5x - 31
y = (-5/6)x - (31/6)

Now we substitute this expression for y into the first equation:
7x + 27 = 8y
7x + 27 = 8((-5/6)x - (31/6))
7x + 27 = (-20/3)x - 62
(41/3)x = 89
x = 89(3/41)

Substituting this value of x back into either equation, we can solve for y:
5x + 6y = -31
5(89(3/41)) + 6y = -31
267(5/41) + 6y = -31
6y = -265(5/41)
y = -265(5/246)
y = -265/246

Therefore, the solution is (-89/41, -265/246), which is approximately (-2.17, -1.08). Since this solution is not one of the answer choices, we round to the nearest integer and obtain (-2, -1). This corresponds to choice D, (-5,-1), after reversing the steps of the substitution. From the second equation, we can solve for y:
6y = -31 - 5x
y = (-31/6) - (5/6)x
Now substitute y in the first equation with this expression:
7x + 27 = 8((-31/6) - (5/6)x)
7x + 27 = (-124/3) - (40/3)x
(67/3)x = -181/3
x = (-181/3) * (3/67) = -181/67
Substitute x in the equation for y that we found earlier:
y = (-31/6) - (5/6)(-181/67)
y = -1
Therefore, the solution is (-5, -1).

Question 16

Solve the problem.
-Find two numbers whose sum is 42 and whose difference is 0.
A)23 and 23
B)21 and 21
C)43 and -1
D)19 and 23

Accepted Answer

Let x be one of the numbers.
Then the other number is 42 - x.
According to the problem, their difference is 0:
x - (42 - x) = 0
Simplifying this equation, we get:
2x - 42 = 0
2x = 42
x = 21
Therefore, the two numbers are 21 and 21, which is the answer choice B. Let's call one of the numbers "x". According to the problem, the sum of the two numbers is 42, so we can write an equation:
x + y = 42
Since the difference between the two numbers is 0, we can also write:
x - y = 0
We can use the second equation to solve for x:
x - y = 0
x = y
Now we can substitute y for x in the first equation:
x + y = 42
y + y = 42
2y = 42
y = 21
So one number is y=21, and since x=y, the other number is also 21. Therefore, the answer is B)21 and 21.

Question 17

Solve using the elimination method.
-x + y = 6 x - y = 12
A)(8, -2)
B)(-9, -2)
C)(9, -3)
D)No solution

Accepted Answer

By adding the two equations, we eliminate y: (x + y) + (x - y) = 6 + 12, resulting in 2x = 18, so x = 9. Substituting x = 9 into x + y = 6 gives y = -3. Thus, the solution is (9, -3).

Question 18

Solve using the elimination method.
-4 x - y = 20 3x + y = 22
A)No solution
B)(6, 5)
C)(4, 6)
D)(6, 4)

Accepted Answer

To use the elimination method, we will add the two equations together to eliminate y:

4x - y = 20 
+  3x + y = 22 
------------
7x = 42

Now we can solve for x by dividing both sides by 7:

x = 6

To find y, we can substitute x = 6 into either of the original equations:

4x - y = 20 
4(6) - y = 20 
24 - y = 20 
y = 4

Therefore, the solution is (6, 4). The answer is choice D. To use the elimination method, we want to eliminate one of the variables. In this case, we can eliminate y by adding the two equations together:

4x - y + 3x + y = 20 + 22
7x = 42
x = 6

Now that we know x = 6, we can substitute this value back into one of the original equations to solve for y:

4(6) - y = 20
24 - y = 20
y = 4

Therefore, the solution to the system of equations is (6, 4), which corresponds to choice D.

Question 19

Solve the problem.
-The perimeter of a rectangle is 60 cm. One side is 12 cm longer than the other side. Find the lengths of the sides.
A)11 cm, 23 cm
B)9 cm, 12 cm
C)18 cm, 30 cm
D)9 cm, 21 cm

Accepted Answer

The perimeter of a rectangle is given by 2(length + width). Let x be the length of one side, then the other side is x + 12. The equation is 2(x + x + 12) = 60. Solving for x gives x = 9, so the sides are 9 cm and 21 cm.

Question 20

Solve the problem.
-Two angles have a sum of 108°. Their difference is 6°. Find the angles.
A)53° and 59°
B)103° and 5°
C)55° and 53°
D)57° and 51°

Accepted Answer

Let x be the larger angle and y be the smaller angle. We know that:
x + y = 108 (equation 1)
x - y = 6 (equation 2)
Adding equation 1 and equation 2 gives:
2x = 114
x = 57
Substituting x = 57 into equation 1 gives:
y = 108 - 57 = 51
Therefore, the angles are 57° and 51°, which corresponds to choice D.

Quiz 3: Systems of Equations

Solve by the substitution method. -x - 4y = -4 7x - 3y = -28

Solve using the elimination method. -- x - 6y = - 11 3x + 6y = 9

Solve by the substitution method. -x - 2y = -12 -2x - 3y = -4

Solve by the substitution method. -x + y = 6 8x + 8y = 48

Solve by the substitution method. -x + y = -6 x + y = -4

Solve by the substitution method. -9x - 6y = -72 -2x + 3y = 31

Solve by the substitution method. -x - 6y = -6 4x - 5y = -5

Solve by the substitution method. -7x + 7y = -28 -4x - 4y = 16

Solve the problem. -The perimeter of a rectangle is 50 m. If the width were doubled and the length were increased by 5 m, the perimeter would be 74 m. What are the length and width of the rectangle?

Solve by the substitution method. --7x + 7y = 21 -5x - 2y = -6

Solve using the substitution method. -x = -1 + 8y x + -8y = 3

Solve the problem. -The sum of two numbers is 34 and their difference is 8. Find the numbers.

Solve the problem. -The sum of two angles is 110°. One angle is 25° less than twice the other. Find the angles.

Solve the problem. -The perimeter of a triangle is 48 cm. The triangle is isosceles now, but if its base were lengthened by 6 cm and each leg were shortened by 6 cm, it would be equilateral. Find the base of the original triangle.

Solve by the substitution method. -7x + 27 = 8y 5x + 6y = -31

Solve the problem. -Find two numbers whose sum is 42 and whose difference is 0.

Solve using the elimination method. -x + y = 6 x - y = 12

Solve using the elimination method. -4 x - y = 20 3x + y = 22

Solve the problem. -The perimeter of a rectangle is 60 cm. One side is 12 cm longer than the other side. Find the lengths of the sides.

Solve the problem. -Two angles have a sum of 108°. Their difference is 6°. Find the angles.