To solve the simultaneous equations using the elimination method, we have the following two equations:
- \( 8y + 4z = 7 \)
- \( 6y - 8z = 41 \)
Step 1: Align the equations for elimination
Let's first manipulate the equations to eliminate one of the variables. We can multiply both equations by suitable constants to make the coefficients of one variable the same.
We'll eliminate \( z \). Since the coefficients of \( z \) in the two equations are \( 4 \) and \( -8 \), we can multiply the first equation by \( 2 \) to match the coefficients of \( z \):
Multiplying the first equation by \( 2 \):
\( 2(8y + 4z) = 2(7) \)
\( 16y + 8z = 14 \) (Equation 3)
Now we have:
- \( 16y + 8z = 14 \)
- \( 6y - 8z = 41 \)
Step 2: Add equations (3) and (4)
Now we can add Equation 3 and Equation 4 together:
\[ (16y + 8z) + (6y - 8z) = 14 + 41 \] \[ 16y + 6y + 8z - 8z = 55 \] \[ 22y = 55 \]
Step 3: Solve for \( y \)
\[ y = \frac{55}{22} \] \[ y = 2.5 \]
Step 4: Substitute \( y \) back into one of the original equations
Now we substitute \( y = 2.5 \) back into one of the original equations. We'll use the first equation:
\[ 8y + 4z = 7 \] Substituting \( y = 2.5 \):
\[ 8(2.5) + 4z = 7 \] \[ 20 + 4z = 7 \] \[ 4z = 7 - 20 \] \[ 4z = -13 \] \[ z = \frac{-13}{4} \] \[ z = -3.25 \]
Final Solution: The solution to the simultaneous equations is:
\[ y = 2.5, \quad z = -3.25 \]
Feel free to ask if you have any follow-up questions!