Use substitution to solve the system of equations given below.
8x - 4y = 16
2x - y = 4
2x - y = 4
Subtract 4 and add y to both sides of the above equation.
y = 2x -4
8x - 4y = 16
Substitute 2x-4 for y in last equation and solve for x. Insert that value into the first equation and solve for y. Check by inserting both values into the second equation.
To solve the system of equations using substitution, we will first solve one equation for one variable and then substitute this expression into the other equation.
Let's solve the second equation for y:
2x - y = 4
Rearranging the equation to isolate y, we get:
y = 2x - 4
Now, substitute this expression for y in the first equation:
8x - 4(2x - 4) = 16
Simplify the equation:
8x - 8x + 16 = 16
Combine like terms:
16 = 16
This equation is true, but it does not provide any information about x. It means that the two original equations are dependent, meaning they are essentially the same line and have an infinite number of solutions.
In conclusion, the system of equations is dependent, and there are infinitely many solutions.
To solve the system of equations using substitution, we'll solve one equation for one variable and substitute it into the other equation.
Let's solve the second equation for y:
2x - y = 4
Adding y to both sides:
2x - y + y = 4 + y
Simplifying:
2x = 4 + y
Now, let's solve the first equation using the value we just found for 2x:
8x - 4y = 16
Substituting 4 + y for 2x:
8(4 + y) - 4y = 16
Expanding:
32 + 8y - 4y = 16
Combining like terms:
32 + 4y = 16
Subtracting 32 from both sides:
4y = 16 - 32
Simplifying:
4y = -16
Dividing by 4:
y = -4
Now that we have the value for y, we can substitute it back into the second equation to find x:
2x - y = 4
Substituting -4 for y:
2x - (-4) = 4
Simplifying:
2x + 4 = 4
Subtracting 4 from both sides:
2x = 4 - 4
Simplifying:
2x = 0
Dividing by 2:
x = 0
Therefore, the solution to the system of equations is x = 0 and y = -4.