In this analysis of systems of equations, we explore how to match equations to their corresponding graphs by utilizing the slope-intercept form, which is expressed as y = mx + b, where m represents the slope and b denotes the y-intercept. This method simplifies the process of identifying the correct graph for each equation.

For the first pair of equations, y = 3x + 5 and y = -2x + 10, we begin by identifying the y-intercepts. The first equation has a y-intercept of 5, while the second has a y-intercept of 10. By examining the graphs, we find that the graph corresponding to y = 3x + 5 intersects the y-axis at 5, confirming its match. Similarly, the graph for y = -2x + 10 intersects at 10, validating our identification.

To further verify the intersection point of the two lines, we substitute the coordinates into both equations. For the point (1, 8), substituting into the first equation yields:

8 = 3(1) + 5

which simplifies to 8 = 8, a true statement. For the second equation:

8 = -2(1) + 10

which also simplifies to 8 = 8, confirming the intersection point is valid.

Next, we analyze the second pair of equations: y = 4x + 8 and another equation with a y-intercept of 10. The first equation has a y-intercept of 8, which we find in one of the graphs. However, since the other graph intersects at 10, we can conclude that the first equation corresponds to the graph with the y-intercept of 8.

Finally, we confirm the last equation, which must align with the graph that has a y-intercept of 10. By rearranging the equations, we can see that both equations indeed have a y-intercept of 10, solidifying their match with the respective graphs.

This systematic approach to matching equations with their graphs not only reinforces understanding of slope-intercept form but also enhances problem-solving skills in identifying intersections and verifying solutions.