Solve the system of equations.

Question

8x−3y= −22
y=10+4x(1 point)

Solve the system of equations.
5x+2y=14
x−6y=22(1 point)

Solve the system of equations.
–9x+y=12
2x+4y= −28(1 point)

How many solutions does the system of equations have?
6x+2y=−18
3x+y=24(1 point)

Answer 1

1. To solve the first system of equations, we will use the substitution method. Since y is already isolated in the second equation, we will substitute it into the first equation:

8x-3(10 + 4x)= -22

Now, we can solve for x:

8x - 30 - 12x = -22

-4x = 8

x = -2

Now substitute this value of x back into the equation for y:

y = 10 + 4(-2)

y = 2

So the solution to the first system is x = -2, y = 2.

2. For the second system, we will use the elimination method. First, multiply the second equation by 2 to make the coefficients of y in both equations the same:

2(x - 6y) = 2(22)

2x - 12y = 44

Now subtract the second equation from the first:

(5x + 2y) - (2x - 12y) = 14 - 44

3x + 14y = -30

Now divide the equation by 3:

x + 14/3 * y = -10

Now we can express y in terms of x and substitute back into the second original equation:

y = 3/14 * x + 10/3

Now substitute this back in the equation x - 6y = 22:

x - 6(3/14 * x + 10/3) = 22

x - 18/14 * x - 20 = 22

-4/14 * x = 42

x = -147

Now substitute this value of x back into the equation for y:

y = 3/14 * (-147) + 10/3

y = -69/2

So the solution to the second system is x = -147, y = -69/2.

3. For the third system, we will use the elimination method. First, multiply the first equation by 4 to make the coefficients of y in both equations the same:

4(-9x + y)= 4(12)

-36x + 4y = 48

Now add the two equations to eliminate y:

-36x + 4y + 2x + 4y = 48 - 28

-34x = 20

Now divide the equation by -34:

x = -10/17

Now substitute this value of x back into the first equation:

-9(-10/17) + y = 12

y = -39/17

So the solution to the third system is x = -10/17, y = -39/17.

4. To find the number of solutions for the last system, we will check for consistency. Divide the first equation by 2:

3x + y = -9

Notice that this equation is the same as the second equation but multiplied by -1. This means that the system has infinitely many solutions, since the two equations represent the same line.