intersect. (f) Solve for \( x \) and \( y \) using the method of your choice. \( 5 x-3 y=10 \) and \( 2 y-3 x=12 \)

To solve the equations \( 5x - 3y = 10 \) and \( 2y - 3x = 12 \), we can use the substitution or elimination method. Let's rearrange the first equation for \( y \): 1. From \( 5x - 3y = 10 \), we can express \( y \) as: \[ 3y = 5x - 10 \quad \Rightarrow \quad y = \frac{5x - 10}{3} \] 2. Substitute this expression for \( y \) into the second equation \( 2y - 3x = 12 \): \[ 2\left(\frac{5x - 10}{3}\right) - 3x = 12 \] 3. Multiply everything by 3 to eliminate the fraction: \[ 2(5x - 10) - 9x = 36 \quad \Rightarrow \quad 10x - 20 - 9x = 36 \quad \Rightarrow \quad x - 20 = 36 \quad \Rightarrow \quad x = 56 \] 4. Substitute \( x = 56 \) back into the equation for \( y \): \[ y = \frac{5(56) - 10}{3} = \frac{280 - 10}{3} = \frac{270}{3} = 90 \] So the solution is \( x = 56 \) and \( y = 90 \). Let's go on to see how these concepts appear throughout history, making our math journey more interesting! In ancient Babylon, around 2000 BC, mathematicians were already solving systems of linear equations, albeit on clay tablets with cuneiform script. They had a sophisticated understanding of these concepts, using methods similar to what we now refer to as the elimination method, demonstrating that the quest for algebraic solutions has deep historical roots! If you want to dive deeper into the world of systems of equations, consider checking out "Elementary Linear Algebra" by Howard Anton. It provides an accessible introduction to linear systems, complete with numerous examples and practice problems that will help strengthen your understanding of how to tackle these equations!