Sure! Let's solve the system of equations using the Elimination method. The given system is:
[tex]\[
\begin{cases}
5x - 4y = 23 & \quad \text{(Equation 1)} \\
7x + 8y = 5 & \quad \text{(Equation 2)}
\end{cases}
\][/tex]
Step-by-Step Solution:
1. Multiply the equations to eliminate [tex]\( y \)[/tex]:
- Multiply Equation 1 by 2 to make the coefficients of [tex]\( y \)[/tex] opposites.
[tex]\[
10x - 8y = 46 \quad \text{(Equation 1 multiplied by 2)}
\][/tex]
- Multiply Equation 2 by 1 (no change needed).
[tex]\[
7x + 8y = 5 \quad \text{(Equation 2 unchanged)}
\][/tex]
2. Add the two equations:
- Adding Equation 1 multiplied by 2 and Equation 2 will eliminate [tex]\( y \)[/tex]:
[tex]\[
(10x - 8y) + (7x + 8y) = 46 + 5
\][/tex]
This simplifies to:
[tex]\[
17x = 51
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
- Divide both sides of the equation by 17:
[tex]\[
x = \frac{51}{17} \implies x = 3
\][/tex]
4. Substitute [tex]\( x \)[/tex] back into one of the original equations:
- Use Equation 1 for substitution:
[tex]\[
5x - 4y = 23
\][/tex]
- Substitute [tex]\( x = 3 \)[/tex] into Equation 1:
[tex]\[
5(3) - 4y = 23
\][/tex]
This simplifies to:
[tex]\[
15 - 4y = 23
\][/tex]
5. Solve for [tex]\( y \)[/tex]:
- Isolate [tex]\( y \)[/tex] by first moving [tex]\( 15 \)[/tex] to the right side:
[tex]\[
-4y = 23 - 15
\][/tex]
This simplifies to:
[tex]\[
-4y = 8
\][/tex]
- Divide both sides by -4:
[tex]\[
y = \frac{8}{-4} \implies y = -2
\][/tex]
So, we have found the solution:
[tex]\[ x = 3, \; y = -2 \][/tex]
6. Conclusion:
The solution to the system of equations is [tex]\( \boxed{(3, -2)} \)[/tex].
Out of the given possible solutions:
- (3, -2)
- (3, 2)
- (-3, -2)
- (-3, 2)
- No Solution
- Infinite Solutions
The correct solution is:
[tex]\[ (3, -2) \][/tex]
