\( 2 x-y=10 \) and \( 3 x+2 y=22 \)

To solve the system of equations \( 2x - y = 10 \) and \( 3x + 2y = 22 \), you can start with the first equation to express \( y \) in terms of \( x \). Rearranging gives \( y = 2x - 10 \). Next, substitute this expression into the second equation: \( 3x + 2(2x - 10) = 22 \). Simplifying, you'll find \( 7x - 20 = 22 \), leading to \( 7x = 42 \) or \( x = 6 \). Substituting \( x \) back to find \( y \) gives \( y = 2(6) - 10 = 2 \). The solution to the system is \( x = 6 \) and \( y = 2 \)! Now that you’ve solved the system, you could visualize it by graphing the two equations. They will intersect at the point (6, 2), which represents the solution. This intersection signifies the only combination of \( x \) and \( y \) that satisfies both conditions at the same time. Graphing these equations can help you gain a clearer understanding of how systems work in geometry! When working with system of equations, be mindful of potential mistakes like misapplying the substitution method or making algebraic errors during calculations. Double-check your arithmetic and be careful when handling negative signs! Also, watch out for declaring a solution too soon; always ensure your solution meets both original equations to confirm it's correct!