Identify the solution(s) of the system of equations, if any.
-2x - 6y = 4
24y = -8x - 16
first equation is
2 x + 6 y = -4
second equation is
2 x + 6 y = -4
the same, no intersecting lines, no solution
To find the solution(s) of the system of equations, we can use the method of substitution.
Let's start by solving the second equation for y:
24y = -8x - 16
Divide both sides of the equation by 24:
y = (-8x - 16) / 24
Simplify the right side of the equation:
y = (-1/3)x - 2/3
Now, substitute this expression for y in the first equation:
-2x - 6((-1/3)x - 2/3) = 4
Distribute the -6 to the terms inside the parentheses:
-2x + 2x/3 + 4/3 = 4
Combine like terms:
(2/3)x + 4/3 = 4
Subtract 4/3 from both sides:
(2/3)x = 4 - 4/3
Simplify the right side of the equation:
(2/3)x = 8/3
To solve for x, multiply both sides of the equation by 3/2:
x = (8/3)(3/2)
Multiply the numerators and denominators:
x = 24/6
Simplify the fraction:
x = 4
Now that we have the value of x, we can substitute it back into either of the original equations to find the value of y.
Using the second equation:
24y = -8(4) - 16
Simplify the right side of the equation:
24y = -32 - 16
Combine like terms:
24y = -48
Divide both sides of the equation by 24:
y = -48/24
Simplify the fraction:
y = -2
So, the solution to the system of equations is x = 4 and y = -2.
To find the solution(s) of the system of equations, we can use the method of substitution or the method of elimination. I will explain how to use the method of substitution.
Step 1: Solve one of the equations for one variable in terms of the other variable.
We'll solve the second equation for x:
24y = -8x - 16
Divide both sides of the equation by -8:
3y = x + 2
Step 2: Substitute the expression for one variable into the other equation.
Now, substitute the expression for x in terms of y into the first equation:
-2x - 6y = 4
-2(3y - 2) - 6y = 4
Simplify the equation:
-6y + 4 - 6y = 4
-12y + 4 = 4
Step 3: Solve the resulting equation for the remaining variable.
Let's solve for y:
-12y = 4 - 4
-12y = 0
Divide both sides of the equation by -12:
y = 0
Step 4: Substitute the value of y back into the equation to solve for x.
Now substitute the value of y back into either of the original equations. Let's use the second equation:
24(0) = -8x - 16
0 = -8x - 16
Add 8x to both sides of the equation:
8x = -16
Divide both sides of the equation by 8:
x = -2
Therefore, the solution to the system of equations is x = -2 and y = 0.