Solve the system of equations [tex]3r - 4s = 0[/tex] and [tex]2r + 5s = 23[/tex].

A. [tex]r = 4, s = 3[/tex]
B. [tex]r = -12, s = 9[/tex]
C. [tex]r = 12, s = 0[/tex]
D. [tex]r = -1, s = -3[/tex]



Answer :

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]