Question Find the solution of the system of equations. \[ \begin{array}{r}5 x-4 y=34 \\ -x+8 y=22\end{array} \] Answer Attempt 1 out of 2

To solve the system of equations, we can use substitution or elimination. Let's convert the first equation into \( y \): From \( 5x - 4y = 34 \): \[ 4y = 5x - 34 \] \[ y = \frac{5x - 34}{4} \] Now substitute this expression for \( y \) into the second equation \( -x + 8y = 22 \): \[ -x + 8\left(\frac{5x - 34}{4}\right) = 22 \] Now, simplify and solve for \( x \): \[ -x + 2(5x - 34) = 22 \] \[ -x + 10x - 68 = 22 \] \[ 9x - 68 = 22 \] \[ 9x = 90 \] \[ x = 10 \] Now, substitute \( x = 10 \) back into the first equation to find \( y \): \[ 5(10) - 4y = 34 \] \[ 50 - 4y = 34 \] \[ -4y = 34 - 50 \] \[ -4y = -16 \] \[ y = 4 \] The solution of the system is \( (x, y) = (10, 4) \). Finding solutions to systems of equations has been a crucial problem throughout history. Ancient mathematicians like the Greeks used geometric methods to solve such problems long before algebra was formalized. The ability to represent and solve these equations mathematically laid the groundwork for modern mathematics and its applications in fields as diverse as engineering and economics. In real-world applications, systems of equations are vital in numerous fields such as economics, engineering, and physics. For example, in business, one may use systems to determine the optimal production levels of different products to maximize profit or minimize costs, considering constraints like material availability or labor hours. Solving these equations accurately can lead to better decision-making and strategic planning!