To solve the given system of equations, we'll equate the corresponding components of the tuples and solve them step by step:
The given tuples are:
[tex]\[
\left(4 x + \frac{3}{y}, 9\right) = \left(7, 3 x + \frac{6}{y}\right)
\][/tex]
From this, we obtain two equations by equating the components:
[tex]\[
4x + \frac{3}{y} = 7 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
9 = 3x + \frac{6}{y} \quad \text{(Equation 2)}
\][/tex]
We'll solve these equations simultaneously.
Step 1: Start with Equation 2:
[tex]\[
9 = 3x + \frac{6}{y}
\][/tex]
Subtract [tex]\(3x\)[/tex] from both sides to isolate [tex]\(\frac{6}{y}\)[/tex]:
[tex]\[
9 - 3x = \frac{6}{y}
\][/tex]
Rewriting the equation, we find:
[tex]\[
\frac{6}{y} = 9 - 3x
\][/tex]
Taking the reciprocal on both sides:
[tex]\[
\frac{y}{6} = \frac{1}{9 - 3x}
\][/tex]
Multiplying both sides by 6 to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{6}{9 - 3x}
\][/tex]
Step 2: Substitute [tex]\(y\)[/tex] from Equation 2 into Equation 1:
[tex]\[
4x + \frac{3}{\frac{6}{9 - 3x}} = 7
\][/tex]
Simplify the fraction:
[tex]\[
4x + \frac{3(9 - 3x)}{6} = 7
\][/tex]
[tex]\[
4x + \frac{27 - 9x}{6} = 7
\][/tex]
Combine the terms over a common denominator:
[tex]\[
4x + \frac{27 - 9x}{6} = 7
\][/tex]
Rewrite [tex]\(4x\)[/tex] with a common denominator:
[tex]\[
\frac{24x}{6} + \frac{27 - 9x}{6} = 7
\][/tex]
Combine the terms in the numerator:
[tex]\[
\frac{24x + 27 - 9x}{6} = 7
\][/tex]
Simplify the numerator:
[tex]\[
\frac{15x + 27}{6} = 7
\][/tex]
Multiply both sides by 6 to clear the fraction:
[tex]\[
15x + 27 = 42
\][/tex]
Subtract 27 from both sides:
[tex]\[
15x = 15
\][/tex]
Divide by 15:
[tex]\[
x = 1
\][/tex]
Step 3: Substitute [tex]\(x = 1\)[/tex] back into the equation [tex]\(y = \frac{6}{9 - 3x}\)[/tex]:
[tex]\[
y = \frac{6}{9 - 3 \cdot 1}
\][/tex]
[tex]\[
y = \frac{6}{6}
\][/tex]
[tex]\[
y = 1
\][/tex]
Thus, the solution to the given system of equations is:
[tex]\[
x = 1 \quad \text{and} \quad y = 1
\][/tex]
