Answer :
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].
[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].
