To solve the system of equations:
[tex]\[
\frac{7}{3} x - \frac{6}{2} y = 15
\][/tex]
[tex]\[
\frac{8}{3} x = \frac{9}{2} y
\][/tex]
we'll first rewrite these in a more standard form.
### Equation 1
Starting with the first equation:
[tex]\[
\frac{7}{3} x - \frac{6}{2} y = 15
\][/tex]
Simplify the coefficients:
[tex]\[
\frac{7}{3} x - 3 y = 15 \quad \text{(since } \frac{6}{2} = 3\text{)}
\][/tex]
### Equation 2
For the second equation:
[tex]\[
\frac{8}{3} x = \frac{9}{2} y
\][/tex]
Rewrite it in a standard form:
[tex]\[
\frac{8}{3} x - \frac{9}{2} y = 0
\][/tex]
Now we have the system:
[tex]\[
\frac{7}{3} x - 3 y = 15
\][/tex]
[tex]\[
\frac{8}{3} x - \frac{9}{2} y = 0
\][/tex]
### Solving the System:
We will solve this system of linear equations using substitution or elimination. In this case, let's isolate [tex]\( x \)[/tex] from the second equation and substitute it into the first equation.
Starting with:
[tex]\[
\frac{8}{3} x = \frac{9}{2} y
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{\frac{9}{2}}{\frac{8}{3}} y = \frac{9}{2} \cdot \frac{3}{8} y = \frac{27}{16} y
\][/tex]
Now substitute [tex]\( x = \frac{27}{16} y \)[/tex] into the first equation:
[tex]\[
\frac{7}{3} \left(\frac{27}{16} y\right) - 3 y = 15
\][/tex]
Simplify the first term:
[tex]\[
\frac{7 \times 27}{3 \times 16} y - 3 y = 15
\][/tex]
[tex]\[
\frac{189}{48} y - 3 y = 15
\][/tex]
Convert [tex]\( 3 y \)[/tex] to a fraction with a denominator of 48:
[tex]\[
\frac{189}{48} y - \frac{144}{48} y = 15
\][/tex]
Combine the fractions:
[tex]\[
\frac{189 - 144}{48} y = 15
\][/tex]
[tex]\[
\frac{45}{48} y = 15
\][/tex]
Reduce the fraction:
[tex]\[
\frac{15}{16} y = 15
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
y = 15 \times \frac{16}{15}
\][/tex]
[tex]\[
y = 16
\][/tex]
Now substitute [tex]\( y = 16 \)[/tex] back into the expression for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{27}{16} y
\][/tex]
[tex]\[
x = \frac{27}{16} \times 16
\][/tex]
[tex]\[
x = 27
\][/tex]
### Solution:
The values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the system of equations are:
[tex]\[
\boxed{x = 27, \, y = 16}
\][/tex]
