To solve a system of equations graphically, we need to identify the intersection point where the lines representing the equations cross. In this example, we are given two equations in slope-intercept form: y = 2x + 3 and y = x + 4.

Starting with the first equation, y = 2x + 3, we can identify the y-intercept as the point (0, 3). The slope, which is 2, indicates that for every 1 unit we move to the right (positive x-direction), we move up 2 units (positive y-direction). Plotting this, we can also move down 2 units and to the left 1 unit to find additional points. Connecting these points will yield a straight line representing the equation.

Next, we analyze the second equation, y = x + 4. Here, the y-intercept is (0, 4) and the slope is 1, meaning we move up 1 unit and over 1 unit to the right. Similarly, we can move down 1 unit and to the left 1 unit to find more points. Drawing this line will show us the graph of the second equation.

Upon graphing both lines, we observe that they intersect at the point (1, 5). This intersection point is crucial as it represents the solution to the system of equations.

To confirm that (1, 5) is indeed a solution to both equations, we substitute the x and y values into each equation. For the first equation, substituting gives us:

y = 2(1) + 3 which simplifies to 5 = 5, a true statement.

For the second equation, substituting yields:

y = 1 + 4 which also simplifies to 5 = 5, confirming it as a true statement.

Since both equations yield true statements when substituting the intersection point, we conclude that (1, 5) is a valid solution to the system of equations.