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

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



Answer :

GinnyAnswer
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]

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