Let's solve the given system of equations step by step.
The system of equations is:
1. [tex]\(3r - 4s = 0\)[/tex]
2. [tex]\(2r + 5s = 23\)[/tex]
### Step 1: Solve the first equation for one variable
We start by solving the first equation for [tex]\(r\)[/tex]:
[tex]\[3r - 4s = 0\][/tex]
Rearrange to solve for [tex]\(r\)[/tex]:
[tex]\[3r = 4s\][/tex]
[tex]\[r = \frac{4s}{3}\][/tex]
### Step 2: Substitute into the second equation
Now, we substitute [tex]\(r = \frac{4s}{3}\)[/tex] into the second equation [tex]\(2r + 5s = 23\)[/tex]:
[tex]\[2\left(\frac{4s}{3}\right) + 5s = 23\][/tex]
Simplify the expression:
[tex]\[\frac{8s}{3} + 5s = 23\][/tex]
Combine the terms to get a common denominator:
[tex]\[\frac{8s + 15s}{3} = 23\][/tex]
[tex]\[ \frac{23s}{3} = 23\][/tex]
### Step 3: Solve for [tex]\(s\)[/tex]
Multiply both sides by 3 to isolate [tex]\(s\)[/tex]:
[tex]\[23s = 69\][/tex]
[tex]\[s = 3\][/tex]
### Step 4: Solve for the other variable [tex]\(r\)[/tex]
Now that we have [tex]\(s = 3\)[/tex], substitute back into [tex]\(r = \frac{4s}{3}\)[/tex]:
[tex]\[r = \frac{4 \cdot 3}{3}\][/tex]
[tex]\[r = 4\][/tex]
### Conclusion
The solution to the system of equations is:
[tex]\[r = 4\][/tex]
[tex]\[s = 3\][/tex]
Thus, the correct answer is:
A. [tex]\(r=4, s=3\)[/tex]
