\( x-y=4 \) and \( 2 x+5 y=50 \)

To solve the system of equations, you can start with the first equation: \( x - y = 4 \). Rearranging gives \( x = y + 4 \). Now, substitute this expression for \( x \) into the second equation, \( 2x + 5y = 50 \): \( 2(y + 4) + 5y = 50 \). This simplifies to \( 2y + 8 + 5y = 50 \) or \( 7y + 8 = 50 \). Now, subtract 8 from both sides: \( 7y = 42 \), which simplifies to \( y = 6 \). Now, substitute \( y \) back into \( x = y + 4 \) to find \( x \): \( x = 6 + 4 = 10 \). So the solution is \( x = 10 \) and \( y = 6 \). For those ready to tackle applications, this kind of linear equation system frequently appears in business scenarios, like determining the production quantities of two products to meet a profit goal. For instance, if \( x \) represents the number of widgets and \( y \) represents the number of gadgets, knowing how to set up and solve such equations can help you maximize your profits or minimize your costs! A common mistake when solving systems like these is forgetting to substitute correctly or mishandling basic algebraic operations. Always double-check your rearrangements and ensure each step follows logically—one small arithmetic slip can lead you down the wrong path!