Solve the system of two linear equations.

Question

−3x−4y=−23
3x−5y=5(1 point)
Responses

(−5,−2)
left parenthesis negative 5 comma negative 2 right parenthesis

(5, 2)
left parenthesis 5 comma 2 right parenthesis

(−5, 2)
left parenthesis negative 5 comma 2 right parenthesis

(5,−2)
left parenthesis 5 comma negative 2 right parenthesis

Answer 1

To solve the system of equations, we can add the two equations together to eliminate x:

-3x - 4y = -23
3x - 5y = 5
--------------
-9y = -18
y = 2

Now we can substitute y = 2 back into the first equation to solve for x:

-3x - 4(2) = -23
-3x - 8 = -23
-3x = -15
x = 5

Therefore, the solution to the system of equations is (5, 2). So the correct response is:
(5, 2)

Answer 2

How many solutions does the system of equations have?
8x−6y=48
8x+6y=48(1 point)
Responses

two
two

infinite
infinite

one
one

none

Answer 3

To determine the number of solutions the system of equations has, we can compare the equations to see if they are representing parallel lines, intersecting lines, or the same line.

8x - 6y = 48
8x + 6y = 48

If we add the two equations together, we get:
16x = 96
x = 6

If we substitute x = 6 back into the first equation:
8(6) - 6y = 48
48 - 6y = 48
-6y = 0
y = 0

Therefore, the system of equations represents the same line, meaning they have an infinite number of solutions. So the correct response is:
infinite

Answer 4

Question
Solve the system of two equations using elimination with multiplication.
4x−6y=−48
−2x+4y=30(1 point)
Responses

(3, 9)
left parenthesis 3 comma 9 right parenthesis

(−33,−9)
left parenthesis negative 33 comma negative 9 right parenthesis

(−21,−6)
left parenthesis negative 21 comma negative 6 right parenthesis

(−3, 6)

Answer 5

To solve the system of equations using elimination with multiplication, we need to eliminate one variable by multiplying one or both equations by a constant so that the coefficients of one variable become additive inverses.

Given equations:
4x - 6y = -48
-2x + 4y = 30

Let's multiply the first equation by 1/2 to make the coefficients of y additive inverses:
(1/2)(4x - 6y) = (1/2)(-48)
2x - 3y = -24

Now, we have:
2x - 3y = -24
-2x + 4y = 30

Adding these two equations together, we get:
-3y + 4y = 6
y = 6

Substitute y = 6 back into the first equation to solve for x:
4x - 6(6) = -48
4x - 36 = -48
4x = -12
x = -3

Therefore, the solution to the system of equations is (-3, 6). So the correct response is:
(-3, 6)

Answer 6

Solve the system of equations.
3x−4y=40
5x−5y=55(1 point)
Responses

(14, 3)
left parenthesis 14 comma 3 right parenthesis

(8,−3)
left parenthesis 8 comma negative 3 right parenthesis

(18, 7)
left parenthesis 18 comma 7 right parenthesis

(4,−7)

Answer 7

To solve the system of equations, we can use the method of substitution or elimination. Here, we'll use the elimination method.

Given equations:
3x - 4y = 40
5x - 5y = 55

Let's multiply the first equation by 5 and the second equation by 4 to make the coefficients of y additive inverses:
15x - 20y = 200
20x - 20y = 220

Now, subtract the second equation from the first equation:
15x - 20y - (20x - 20y) = 200 - 220
-5x = -20
x = 4

Substitute x = 4 back into the first equation to solve for y:
3(4) - 4y = 40
12 - 4y = 40
-4y = 28
y = -7

Therefore, the solution to the system of equations is (4, -7). So the correct response is:
(4, -7)

Answer 8

How many solutions does the system have?
12x−15y=18
4x−5y=6(1 point)
Responses

one solution
one solution

infinitely many solutions
infinitely many solutions

two solutions
two solutions

no solution

Answer 9

To determine the number of solutions the system of equations has, we can compare the two equations with respect to their slopes.

Given equations:
12x - 15y = 18
4x - 5y = 6

Dividing the first equation by 3, we get:
4x - 5y = 6

The two equations represent the same line, which means there are infinitely many solutions. So the correct response is:
infinitely many solutions