Question Find the solution of the system of equations. \[ \begin{array}{r}8 x-3 y=2 \\ -8 x-5 y=46\end{array} \] Answer Attempt 1 out of 2

To solve the system of equations, we can start by isolating one of the variables. Let’s take the first equation \(8x - 3y = 2\) and express \(y\) in terms of \(x\): \[ 3y = 8x - 2 \implies y = \frac{8x - 2}{3} \] We can substitute this expression for \(y\) into the second equation \(-8x - 5y = 46\): \[ -8x - 5\left(\frac{8x - 2}{3}\right) = 46 \] Now, let's clear the fraction by multiplying through by 3: \[ -24x - 5(8x - 2) = 138 \] Expanding this gives: \[ -24x - 40x + 10 = 138 \implies -64x + 10 = 138 \] Then, isolate \(x\): \[ -64x = 138 - 10 \implies -64x = 128 \implies x = -2 \] Now, substitute \(x = -2\) back into the expression for \(y\): \[ y = \frac{8(-2) - 2}{3} = \frac{-16 - 2}{3} = \frac{-18}{3} = -6 \] Thus, the solution to the system of equations is \((x, y) = (-2, -6)\). 💡 Here's a fun bit for you: The method we used is often akin to “substitution magic”! It’s like solving a mystery where you find one clue (variable) that leads you to another, ultimately uncovering the hidden secret (solution)! And speaking of mysteries, real-life applications of linear systems like this pop up everywhere – from budgeting your finances to understanding supply and demand in markets! Companies rely on these equations to decide how much of a product to produce based on different factors. It's all in a day's work for most businesses!