Solving Equations Using Substitution Quick Check
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Question
Solve the system of two equations using substitution.
y=2+3x
4x−3y=−16 (1 point)
Responses
(−2, −4)
left parenthesis negative 2 comma negative 4 right parenthesis
(2, 8)
left parenthesis 2 comma 8 right parenthesis
(2, 10)
left parenthesis 2 comma 10 right parenthesis
(−2, 4)
left parenthesis negative 2 comma 4 right parenthesis
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To solve the system of equations using substitution, we can start by substituting the first equation into the second equation:
y = 2 + 3x
4x - 3(2 + 3x) = -16
4x - 6 - 9x = -16
-5x - 6 = -16
-5x = -10
x = 2
Now that we have found the value of x, we can substitute it back into the first equation to find the value of y:
y = 2 + 3(2)
y = 2 + 6
y = 8
Therefore, the solution to the system of equations is (2, 8). So the correct response is:
(2, 8)
left parenthesis 2 comma 8 right parenthesis
Question
Solve the system of equations.
2x+6y=−18
x=5y−1(1 point)
Responses
(−7.25, −1.25)
left parenthesis negative 7.25 comma negative 1.25 right parenthesis
(−6, −1)
left parenthesis negative 6 comma negative 1 right parenthesis
(5.25, 1.25)
left parenthesis 5.25 comma 1.25 right parenthesis
(4, 1)
To solve the system of equations, we'll first substitute the expression for x from the second equation into the first equation:
2(5y - 1) + 6y = -18
10y - 2 + 6y = -18
16y - 2 = -18
16y = -16
y = -1
Now that we have found the value of y, we can substitute it back into the second equation to find the value of x:
x = 5(-1) - 1
x = -5 - 1
x = -6
Therefore, the solution to the system of equations is (-6, -1). So the correct response is:
(-6, -1)
left parenthesis negative 6 comma negative 1 right parenthesis
Solve the system of equations.
3x+y=2
7x−4y=30(1 point)
Responses
(2, −4)
left parenthesis 2 comma negative 4 right parenthesis
(12, 12)
left parenthesis start fraction 1 over 2 end fraction comma start fraction 1 over 2 end fraction right parenthesis
(−2, 8)
left parenthesis negative 2 comma 8 right parenthesis
(2, −2)
To solve the system of equations, we'll use substitution.
First, solve one of the equations for one of the variables. Let's solve the first equation for y:
3x + y = 2
y = 2 - 3x
Now substitute this expression for y into the second equation:
7x - 4(2 - 3x) = 30
7x - 8 + 12x = 30
19x - 8 = 30
19x = 38
x = 2
Now that we have found the value of x, we can substitute it back into the first equation to find the value of y:
3(2) + y = 2
6 + y = 2
y = -4
Therefore, the solution to the system of equations is (2, -4). So the correct response is:
(2, -4)
left parenthesis 2 comma -4 right parenthesis
Solve the system of equations.
−4x+6y=38
x−5y=−20(1 point)
Responses
(−35, −3)
left parenthesis negative 35 comma negative 3 right parenthesis
(35, −3)
left parenthesis 35 comma negative 3 right parenthesis
(−5, 3)
left parenthesis negative 5 comma 3 right parenthesis
(5, 3)
left parenthesis 5 comma 3 right parenthesis
To solve the system of equations, let's use substitution. We can start by solving the second equation for x:
x - 5y = -20
x = 5y - 20
Now substitute this expression for x into the first equation:
-4(5y - 20) + 6y = 38
-20y + 80 + 6y = 38
-14y + 80 = 38
-14y = -42
y = 3
Now that we have found the value of y, we can substitute it back into the second equation to find the value of x:
x = 5(3) - 20
x = 15 - 20
x = -5
Therefore, the solution to the system of equations is (-5, 3). So the correct response is:
(-5, 3)
left parenthesis -5 comma 3 right parenthesis
How many solution(s) does the system of equations have?
−5x+y=−18
10x−2y=36(1 point)
Responses
one
one
two
two
infinite
infinite
none
To determine the number of solutions for the system of equations, we can calculate the slope of both equations. If the slopes are different, there is one solution. If the slopes are the same and the y-intercepts are different, there are no solutions.
Let's rewrite the first equation as y = 5x - 18. This has a slope of 5.
Now let's rewrite the second equation as y = 5x - 18. This also has a slope of 5.
Since both equations have the same slope and the same y-intercept, they represent the same line. The system of equations has infinite solutions.
Therefore, the correct response is:
infinite
infinite