Let's solve the system of equations:
[tex]\[
\begin{cases}
12y + 15x = 21 \\
5y - 3x = 85
\end{cases}
\][/tex]
1. Equation Setup:
- Equation 1: [tex]\(12y + 15x = 21\)[/tex]
- Equation 2: [tex]\(5y - 3x = 85\)[/tex]
2. Express One Variable:
Let's solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. We begin by solving one of the equations for one variable, and then substituting that into the other equation. However, for simplicity and accuracy, we directly aim to solve it using an elimination method or substitution method.
3. Elimination Method:
To eliminate one of the variables, we need to make the coefficients of either [tex]\(x\)[/tex] or [tex]\(y\)[/tex] the same in both equations.
Notice that we can eliminate [tex]\(x\)[/tex] by manipulating the equations.
4. Make the Coefficients Match:
- Multiply Equation 1 by 3 and Equation 2 by 15, so the coefficients of [tex]\(x\)[/tex] in both equations will be the same (45):
[tex]\[
3(12y + 15x) = 3(21) \implies 36y + 45x = 63
\][/tex]
[tex]\[
15(5y - 3x) = 15(85) \implies 75y - 45x = 1275
\][/tex]
5. Add the Equations:
Add the two new equations together:
[tex]\[
(36y + 45x) + (75y - 45x) = 63 + 1275
\][/tex]
[tex]\[
36y + 75y = 1338
\][/tex]
[tex]\[
111y = 1338
\][/tex]
6. Solve for [tex]\(y\)[/tex]:
Divide both sides by 111:
[tex]\[
y = \frac{1338}{111}
\][/tex]
Simplify the fraction:
[tex]\[
y = \frac{1338 \div 3}{111 \div 3} = \frac{446}{37}
\][/tex]
7. Substitute [tex]\(y\)[/tex] back into one of the original equations:
Use Equation 1 for substitution:
[tex]\[
12y + 15x = 21
\][/tex]
Substitute [tex]\(y = \frac{446}{37}\)[/tex]:
[tex]\[
12\left(\frac{446}{37}\right) + 15x = 21
\][/tex]
8. Simplify and solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{12 \cdot 446}{37} + 15x = 21
\][/tex]
[tex]\[
\frac{5352}{37} + 15x = 21
\][/tex]
Subtract [tex]\(\frac{5352}{37}\)[/tex] from both sides:
[tex]\[
15x = 21 - \frac{5352}{37}
\][/tex]
Convert 21 to a fraction with a denominator of 37:
[tex]\[
21 = \frac{777}{37}
\][/tex]
[tex]\[
15x = \frac{777}{37} - \frac{5352}{37}
\][/tex]
[tex]\[
15x = \frac{777 - 5352}{37}
\][/tex]
[tex]\[
15x = \frac{-4575}{37}
\][/tex]
Divide both sides by 15:
[tex]\[
x = \frac{-4575}{37 \cdot 15}
\][/tex]
Simplify:
[tex]\[
x = \frac{-4575}{555}
\][/tex]
[tex]\[
x = \frac{-305}{37}
\][/tex]
So, the solution to the system of equations is:
[tex]\[
\boxed{x = \frac{-305}{37}, \; y = \frac{446}{37}}
\][/tex]
