To solve the system of equations given:
[tex]\[3r - 4s = 0 \quad \text{(Equation 1)}\][/tex]
[tex]\[2r + 5s = 23 \quad \text{(Equation 2)}\][/tex]
Let's go through the steps to find the values of [tex]\(r\)[/tex] and [tex]\(s\)[/tex].
Step 1: Solve Equation 1 for [tex]\(r\)[/tex].
[tex]\[3r - 4s = 0\][/tex]
[tex]\[3r = 4s\][/tex]
[tex]\[r = \frac{4s}{3} \quad \text{(Equation 3)}\][/tex]
Step 2: Substitute Equation 3 into Equation 2.
[tex]\[2r + 5s = 23\][/tex]
[tex]\[2\left(\frac{4s}{3}\right) + 5s = 23\][/tex]
[tex]\[\frac{8s}{3} + 5s = 23\][/tex]
Step 3: Combine like terms.
First, find a common denominator for the fractions:
[tex]\[\frac{8s}{3} + \frac{15s}{3} = 23\][/tex]
[tex]\[\frac{8s + 15s}{3} = 23\][/tex]
[tex]\[\frac{23s}{3} = 23\][/tex]
Step 4: Solve for [tex]\(s\)[/tex].
Multiply both sides by 3 to clear the fraction:
[tex]\[23s = 69\][/tex]
[tex]\[s = \frac{69}{23}\][/tex]
[tex]\[s = 3\][/tex]
Step 5: Substitute [tex]\(s = 3\)[/tex] back into Equation 3 to find [tex]\(r\)[/tex].
[tex]\[r = \frac{4s}{3}\][/tex]
[tex]\[r = \frac{4 \cdot 3}{3}\][/tex]
[tex]\[r = 4\][/tex]
So, the solution to the system of equations is:
[tex]\[r = 4, \quad s = 3\][/tex]
Thus, the correct answer is:
B. [tex]\(r = 4, s = 3\)[/tex]
