To determine the correct graph for the system of equations, we first need to rearrange both equations into the standard format of y = mx + b, where m is the slope and b is the y-intercept.
1. **First Equation:** y = 2x + 1
Here, the slope (m) is 2, and the y-intercept (b) is 1. This means that the line will cross the y-axis at (0, 1), and for every unit increase in x, y increases by 2.
2. **Second Equation:** 3y = x + 4
To put this in y = mx + b form, we can divide the entire equation by 3:
y = (1/3)x + (4/3)
In this case, the slope is 1/3, and the y-intercept is 4/3. This line will cross the y-axis at (0, 4/3), and for every unit increase in x, y increases by 1/3.
Now that we have both equations in slope-intercept form, we can analyze the key points:
- Graph of the first equation (y = 2x + 1) will be steep, starting at (0, 1).
- Graph of the second equation (y = (1/3)x + (4/3)) will be much gentler, starting at (0, 4/3).
To graph these lines accurately:
- Plot the point (0, 1) for the first equation and draw a line with a steep incline.
- Plot the point (0, 4/3) for the second equation and draw a line with a gentle incline.
Finally, the intersection point of these two lines will represent the solution to the system of equations. When both lines are graphed on the same coordinate plane, you can see how they intersect and create a visual representation of where the solution lies.