To solve the system of equations using substitution, we first need to express one variable in terms of the other from one of the equations.
We start with the second equation:
\[ 4x + y = 11 \]
We can solve for \( y \):
\[ y = 11 - 4x \]
Now we can substitute this expression for \( y \) into the first equation:
\[ 8x + 4y = 4 \]
Substituting \( y \):
\[ 8x + 4(11 - 4x) = 4 \]
Now, simplify the equation:
\[ 8x + 44 - 16x = 4 \]
Combine like terms:
\[ -8x + 44 = 4 \]
Next, isolate \( x \):
\[ -8x = 4 - 44 \] \[ -8x = -40 \]
Now, divide by -8:
\[ x = 5 \]
Now that we have \( x \), we can substitute it back into the expression we found for \( y \):
\[ y = 11 - 4x \] \[ y = 11 - 4(5) \] \[ y = 11 - 20 \] \[ y = -9 \]
Thus, the solution to the system of equations is:
\[ \boxed{(5, -9)} \]
Verification:
We can verify our solution by substituting \( (5, -9) \) back into the original equations.
-
For the first equation \( 8x + 4y = 4 \): \[ 8(5) + 4(-9) = 40 - 36 = 4 \quad \text{(True)} \]
-
For the second equation \( 4x + y = 11 \): \[ 4(5) + (-9) = 20 - 9 = 11 \quad \text{(True)} \]
Both equations are satisfied, confirming that the solution \( (5, -9) \) is correct.
1 answer