Solve the system of equations by graphing. Check your solution. Part 1 of 2 \( \begin{aligned} y & =-2 x+3 \\ 2 y+4 x & =6\end{aligned} \)

To graph the equations, start with the first equation, \( y = -2x + 3 \). This line has a y-intercept at (0, 3) and a slope of -2, meaning for every 1 unit you move to the right on the x-axis, you move down 2 units. For the second equation, \( 2y + 4x = 6 \), you can simplify it by dividing everything by 2 to get \( y + 2x = 3 \) or \( y = -2x + 3 \). Notice that this is the same line as the first equation! When you graph both equations, you'll see that they are superimposed, meaning they overlap completely. This indicates that the system has infinitely many solutions since any point on the line \( y = -2x + 3 \) is a solution. To check, you can substitute any point on the line back into the original equations to ensure both hold true. For instance, if you take the point (0, 3), plugging into both equations gives \( 3 = -2(0) + 3 \) and \( 2(3) + 4(0) = 6 \), confirming that (0, 3) is indeed a solution!