To determine which set of coordinates satisfies the given system of equations [tex]\(3x - 2y = 15\)[/tex] and [tex]\(4x - y = 20\)[/tex], we follow these steps:
1. Identify the System of Linear Equations:
[tex]\[
3x - 2y = 15 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
4x - y = 20 \quad \text{(Equation 2)}
\][/tex]
2. Substitute the Possible Solutions into the Equations:
- Option A: [tex]\((2, -7)\)[/tex]
[tex]\[
\text{Check Equation 1:} \quad 3(2) - 2(-7) = 6 + 14 = 20 \quad (\text{Not } 15)
\][/tex]
[tex]\[
\text{Check Equation 2:} \quad 4(2) - (-7) = 8 + 7 = 15 \quad (\text{Not } 20)
\][/tex]
- Option B: [tex]\((1, 1)\)[/tex]
[tex]\[
\text{Check Equation 1:} \quad 3(1) - 2(1) = 3 - 2 = 1 \quad (\text{Not } 15)
\][/tex]
[tex]\[
\text{Check Equation 2:} \quad 4(1) - 1 = 4 - 1 = 3 \quad (\text{Not } 20)
\][/tex]
- Option C: [tex]\((5, 0)\)[/tex]
[tex]\[
\text{Check Equation 1:} \quad 3(5) - 2(0) = 15 - 0 = 15 \quad (\text{True})
\][/tex]
[tex]\[
\text{Check Equation 2:} \quad 4(5) - 0 = 20 - 0 = 20 \quad (\text{True})
\][/tex]
- Option D: [tex]\((0, -7.5)\)[/tex]
[tex]\[
\text{Check Equation 1:} \quad 3(0) - 2(-7.5) = 0 + 15 = 15 \quad (\text{True})
\][/tex]
[tex]\[
\text{Check Equation 2:} \quad 4(0) - (-7.5) = 0 + 7.5 = 7.5 \quad (\text{Not } 20)
\][/tex]
3. Determine the Satisfying Solution:
- After checking each option, we see that only option C [tex]\((5, 0)\)[/tex] satisfies both equations.
So, the correct set of coordinates that satisfy the equations [tex]\(3x - 2y = 15\)[/tex] and [tex]\(4x - y = 20\)[/tex] is:
[tex]\[ \boxed{(5, 0)} \][/tex]
