4-22. When you write equations to solve ward problerns. you sometimes end up with twe equations like Renard's, or like the two equations at right. Notice that the socond equat is solved for \( y \), but the first is not. Change the first equation im:0 \( 7^{-1} \) form and thon \[ \begin{array}{l} x-2 y=4 \\ y=-\frac{1}{2} x+4 \end{array} \] solve this system of equations. Check your solution.

To change the first equation \( x - 2y = 4 \) into slope-intercept form (\( y = mx + b \)), we can solve for \( y \). Start by isolating \( 2y \): \[ 2y = x - 4 \] Now, divide by \( 2 \) to find \( y \): \[ y = \frac{1}{2}x - 2 \] Now our system looks like this: \[ \begin{array}{l} y = \frac{1}{2}x - 2 \\ y = -\frac{1}{2}x + 4 \end{array} \] To solve the system, set the equations equal to each other: \[ \frac{1}{2}x - 2 = -\frac{1}{2}x + 4 \] Combine terms: \[ x + 2 = 4 \] Thus, \( x = 2 \). Substitute \( x = 2 \) back into either equation to find \( y \): \[ y = \frac{1}{2}(2) - 2 = 1 - 2 = -1 \] So, the solution is \( (2, -1) \). Checking, substituting into the original equations confirms that both hold true! Now, let’s celebrate our solution! Not only did we solve those equations, but we also performed mathematical gymnastics to ensure everything balanced out correctly. Next time you face a similar problem, remember to have fun with the process—like a dance where every equation gets its moment in the spotlight!