Solve using the substitution method. Show your work. If the system has no solution or an infinite number of solutions, state this.
8y + 2x = 48
2x = 4y – 12
from the second:
x = 2y - 6
sub into the first
8y + 2(2y-6) = 48
8y + 4y - 12 = 48
12y = 60
y = 5
into the second:
x = 2(5) - 6
= 4
To solve this system of equations using the substitution method, we will solve one equation for one variable and then substitute it into the other equation.
Let's solve the second equation for x:
2x = 4y – 12
Divide both sides of the equation by 2 to isolate x:
x = 2y – 6
Now that we have x in terms of y, we can substitute this expression into the first equation:
8y + 2x = 48
Replace x with 2y – 6:
8y + 2(2y – 6) = 48
Simplify:
8y + 4y – 12 = 48
12y – 12 = 48
Add 12 to both sides of the equation:
12y = 48 + 12
12y = 60
Divide both sides by 12:
y = 5
Now that we have the value of y, we can substitute it back into the equation x = 2y – 6 to find x:
x = 2(5) – 6
x = 10 – 6
x = 4
So the solution to the system of equations is x = 4 and y = 5.
Therefore, the substitution method shows that the system has only one unique solution: x = 4, y = 5.