To solve the system of equations by graphing, we will analyze both equations separately and then identify their point of intersection, which represents the solution to the system.
Step 1: Rearrange the equations
The given equations are:
- Equation 1: y = 3x + 1
- Equation 2: y = x + 7
Both equations are already in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Step 2: Identify the y-intercepts
The y-intercept of an equation is the point at which the line crosses the y-axis (where x = 0).
- For y = 3x + 1, the y-intercept is (0, 1).
- For y = x + 7, the y-intercept is (0, 7).
Step 3: Determine additional points
It’s helpful to plot additional points for each equation:
- For y = 3x + 1, let’s find y when:
- x = 1: y = 3(1) + 1 = 4 → (1, 4)
- x = -1: y = 3(-1) + 1 = -2 → (-1, -2)
- For y = x + 7, let’s find y when:
- x = 1: y = 1 + 7 = 8 → (1, 8)
- x = -1: y = -1 + 7 = 6 → (-1, 6)
Step 4: Plotting the points
On a graph:
- Plot (0, 1) and (1, 4) for the first equation and mark the points.
- Plot (0, 7) and (1, 8) for the second equation and mark those points.
Draw straight lines through the plotted points for each equation to visualize where they intersect.
Step 5: Finding the point of intersection
The point where the two lines cross is the solution to the system of equations. You can see from the graph that:
- Using a graphing tool, you will eventually find that the lines intersect at the point (2, 7).
Final Answer
Therefore, the solution to the system of equations y = 3x + 1 and y = x + 7 is (2, 7).