To solve the given equations graphically, we will first rewrite each equation in a slope-intercept form (y = mx + b) for easier graphing.
Step 1: Rearranging the Equations
1. Starting with the first equation: x + 3y = 6
Rearranging to solve for y:
3y = 6 – x
y = - rac{1}{3}x + 2
2. Now, for the second equation: 2x + 3y = 12
Rearranging to solve for y:
3y = 12 – 2x
y = - rac{2}{3}x + 4
Step 2: Plotting the Equations
Now that we have both equations in slope-intercept form, we can draw them on a graph:
- For the equation y = - rac{1}{3}x + 2, the slope is - rac{1}{3}, and the y-intercept is 2. This means you start at the point (0, 2) on the graph, and for each unit you move to the right, you move down one third of a unit.
- For the equation y = - rac{2}{3}x + 4, the slope is - rac{2}{3}, and the y-intercept is 4. Starting at (0, 4), for each unit you move to the right, you move down two thirds of a unit.
Step 3: Identifying Intersections
After plotting both lines on the graph, look for the point where they intersect. The x and y coordinates of this intersection represent the solution to the system of equations.
Step 4: Verifying the Solution
Once you find the intersection point, substitute the x and y values back into the original equations to verify that both equations are satisfied with this point.
By following these steps, you can efficiently solve the equations graphically and understand the relationship between them visually.