Choosing the Correct Graph for the System of Equations
To identify the correct graph for the given system of equations, we need to understand each equation and find their graphical representation.
Step 1: Rewriting the Equations
The two equations are:
- Equation 1: y = 2x + 1
- Equation 2: 3y = x + 4
Equation 1: y = 2x + 1
This equation is already in slope-intercept form (y = mx + b) where:
- slope (m) = 2
- y-intercept (b) = 1
To graph this line, start at the y-intercept (0,1) and use the slope to find another point. The slope of 2 means that for every 1 unit you move to the right (positive direction along the x-axis), you move up 2 units on the y-axis. The second point will be at (1, 3).
Equation 2: 3y = x + 4
First, we should convert this into slope-intercept form. Divide both sides by 3:
y = (1/3)x + (4/3)
In this case:
- slope (m) = 1/3
- y-intercept (b) = 4/3
Start at the y-intercept (0, 4/3) which is approximately (0, 1.33). From there, because the slope is 1/3, for every 3 units you move to the right, you move up 1 unit. You can plot another point at (3, 2).
Step 2: Plotting the Graphs
Now that we have the points for both equations, you can plot them on the same set of axes:
- For y = 2x + 1, plot points (0, 1) and (1, 3).
- For y = (1/3)x + (4/3), plot points (0, 4/3) and (3, 2).
Step 3: Finding the Intersection
The solution to the system of equations is the point where the two lines intersect. Graphing both lines will clearly show you the intersection point, which is the solution to both equations (x, y).
Conclusion
By graphing these equations and finding their intersection, you can choose the correct graph among multiple options provided. Remember to check for accuracy in both equations and confirm by substitution or additional calculation if necessary!