Solve the system of two equations using substitution.

Question

y=2+3x
4x−3y=−16 (1 point)
___________________________________
2.) Solve the system of equations.
2x+6y=−18
x=5y−1 (1 point)
___________________________________3.) Solve the system of equations.
3x+y=2
7x−4y=30 (1 point)
___________________________________4.)Solve the system of equations.
−4x+6y=38
x−5y=−20 (1 point)
___________________________________5.) How many solution(s) does the system of equations have?
−5x+y=−18
10x−2y=36 (1 point)

Answer 1

1.) To solve this system of equations using substitution, we can start by rearranging the first equation to solve for x:

y = 2 + 3x
3x = y - 2
x = (y - 2)/3

Now we can substitute this expression for x into the second equation:

4x - 3y = -16
4((y - 2)/3) - 3y = -16
(4y - 8)/3 - 3y = -16
4y - 8 - 9y = -48
-5y = -40
y = 8

Substituting this value of y back into the first equation gives:

x = (8 - 2)/3
x = 2

Therefore, the solution to the system of equations is x = 2 and y = 8.

2.) To solve this system of equations, we can substitute the expression for x into the first equation:

2x + 6y = -18
2(5y - 1) + 6y = -18
10y - 2 + 6y = -18
16y = -16
y = -1

Substituting this value of y back into the second equation gives:

x = 5(-1) - 1
x = -5 - 1
x = -6

Therefore, the solution to the system of equations is x = -6 and y = -1.

3.) To solve this system of equations using substitution, we can start by solving the second equation for x:

7x - 4y = 30
7x = 4y + 30
x = (4y + 30)/7

Now we can substitute this expression for x into the first equation:

3x + y = 2
3((4y + 30)/7) + y = 2
(12y + 90)/7 + y = 2
12y + 90 + 7y = 14
19y + 90 = 14
19y = -76
y = -4

Substituting this value of y back into the second equation gives:

x = (4(-4) + 30)/7
x = (-16 + 30)/7
x = 14/7
x = 2

Therefore, the solution to the system of equations is x = 2 and y = -4.

4.) To solve this system of equations, we can start by rearranging the second equation to solve for x:

x - 5y = -20
x = 5y - 20

Now we can substitute this expression for x into the first equation:

-4x + 6y = 38
-4(5y - 20) + 6y = 38
-20y + 80 + 6y = 38
-14y = -42
y = 3

Substituting this value of y back into the second equation gives:

x = 5(3) - 20
x = 15 - 20
x = -5

Therefore, the solution to the system of equations is x = -5 and y = 3.

5.) To determine the number of solutions for this system of equations, we can rearrange Equation 1 to solve for y:

-5x + y = -18
y = 5x - 18

Now let's compare this expression for y to the second equation:

10x - 2y = 36

Since we can solve for y in terms of x using Equation 1, we can substitute this expression into Equation 2:

10x - 2(5x - 18) = 36
10x - 10x + 36 = 36
36 = 36

Since the equation is always true (36 = 36), this means that the two equations represent the same line and they have infinitely many solutions.

Therefore, the system of equations has infinitely many solutions.