To determine which ordered pair [tex]\((r, s)\)[/tex] is the solution to the system of equations:
[tex]\[
\begin{cases}
5r + 7s = 61 \\
-5r + 7s = -19
\end{cases}
\][/tex]
we need to solve the system step-by-step.
1. Add the Equations:
[tex]\[
(5r + 7s) + (-5r + 7s) = 61 + (-19)
\][/tex]
This simplifies to:
[tex]\[
5r - 5r + 7s + 7s = 61 - 19
\][/tex]
[tex]\[
14s = 42
\][/tex]
Solving for [tex]\(s\)[/tex]:
[tex]\[
s = \frac{42}{14} = 3
\][/tex]
2. Substitute [tex]\(s = 3\)[/tex] into the First Equation:
[tex]\[
5r + 7(3) = 61
\][/tex]
Simplifying,
[tex]\[
5r + 21 = 61
\][/tex]
[tex]\[
5r = 61 - 21
\][/tex]
[tex]\[
5r = 40
\][/tex]
Solving for [tex]\(r\)[/tex]:
[tex]\[
r = \frac{40}{5} = 8
\][/tex]
3. Solution Verification:
To ensure the solution is correct, substitute [tex]\(r = 8\)[/tex] and [tex]\(s = 3\)[/tex] into the second equation:
[tex]\[
-5(8) + 7(3) = -19
\][/tex]
Simplifying,
[tex]\[
-40 + 21 = -19
\][/tex]
The equation holds true.
Therefore, the solution to the system of equations is the ordered pair [tex]\((r, s) = (8, 3)\)[/tex].
To check which of the given ordered pairs fits our solution:
- [tex]\((1,8)\)[/tex] does not match [tex]\((8, 3)\)[/tex].
- [tex]\((3,8)\)[/tex] does not match [tex]\((8, 3)\)[/tex].
- [tex]\((1,-2)\)[/tex] does not match [tex]\((8, 3)\)[/tex].
- [tex]\((8,3)\)[/tex] matches perfectly.
Thus, the ordered pair [tex]\((r, s)\)[/tex] that is the solution to the given system of equations is [tex]\((8, 3)\)[/tex].
