Solve the system of equations.
2x + 6y = −18
x = 5y − 1
(1 point)
Responses
(5.25, 1.25)
(−6, −1)
(4, 1)
(−7.25, −1.25)
To solve the system of equations, we can substitute the value of 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, substituting the value of y back into the second equation to find x:
x = 5(-1) - 1
x = -5 - 1
x = -6
Therefore, the solution to the system of equations is (x, y) = (-6, -1).
The correct response is: (−6, −1)
Are you correct
Yes, the correct solution to the system of equations is (−6, −1).
Solve the system of equations. 3x+y=2 7x−4y=30 (1 point) Responses (2, −2) (−2, 8) (1/2, 1/2) (2, −4)
To solve the system of equations, we can use the method of substitution or elimination.
We'll use the method of substitution:
From the first equation, we can express y in terms of x:
y = 2 - 3x
Now, substitute this value of y into the second equation:
7x - 4(2 - 3x) = 30
7x - 8 + 12x = 30
19x - 8 = 30
19x = 38
x = 2
Now, substitute the value of x back into the first equation to solve for y:
3(2) + y = 2
6 + y = 2
y = -4
Therefore, the solution to the system of equations is (x, y) = (2, -4).
The correct response is: (2, -4).
Solve the system of equations. −4x+6y=38 x−5y=−20 (1 point) Responses (5, 3) (35, −3) (−5, 3) (−35, −3)
To solve the system of equations, we can use the method of substitution or elimination.
We'll use the method of substitution:
From the second equation, we can express x in terms of y:
x = 5y - 20
Now, substitute this value of x into the first equation:
-4(5y - 20) + 6y = 38
-20y + 80 + 6y = 38
-14y + 80 = 38
-14y = -42
y = 3
Now, substitute the value of y back into the second equation to solve for x:
x - 5(3) = -20
x - 15 = -20
x = -5
Therefore, the solution to the system of equations is (x, y) = (-5, 3).
The correct response is: (-5, 3).
Are you sure
Apologies for the confusion. Let's solve the system of equations again.
To solve the system of equations, we'll use the method of elimination.
Multiply the second equation by 4 to get:
4(x - 5y) = 4(-20)
4x - 20y = -80
Now, we have the following system of equations:
-4x + 6y = 38
4x - 20y = -80
Adding the equations, we eliminate x:
-4x + 6y + 4x - 20y = 38 - 80
-14y = -42
y = -42 / -14
y = 3
Substitute this value of y back into the second equation to solve for x:
x - 5(3) = -20
x - 15 = -20
x = -20 + 15
x = -5
Therefore, the solution to the system of equations is (x, y) = (-5, 3).
The correct response is: (-5, 3).
How many solution(s) does the system of equations have?
−5x+y=−18
10x−2y=36
(1 point) Responses
none
two
one
infinite
To determine the number of solutions for the system of equations, we can use the method of elimination.
Multiplying the first equation by 2, we get:
-10x + 2y = -36
Now, we have the following system of equations:
-5x + y = -18
-10x + 2y = -36
Notice that the second equation is simply twice the first equation. This implies that the two equations represent the same line and are thus dependent on each other.
When two equations represent the same line, they have infinitely many solutions.
Therefore, the system of equations has infinite solutions.
The correct response is: infinite.
Is this true?
Apologies for the incorrect response. Allow me to correct the answer.
To determine the number of solutions for the system of equations, we can compare the coefficients of x and y in both equations.
Equation 1: -5x + y = -18
Equation 2: 10x - 2y = 36
If we multiply Equation 1 by -2, we get:
10x - 2y = 36
We can see that Equation 2 and -2 times Equation 1 are the same equation. This means that the two equations represent the same line and are dependent on each other.
When two equations are dependent, they have infinitely many solutions. Every point on the line of the equation satisfies the system of equations.
Therefore, the system of equations has infinitely many solutions.
The correct response is: infinite.