To solve the given system of equations graphically, we need to plot both equations on the same set of axes and identify the point where they intersect. Below are step-by-step instructions on how to graph these equations manually.
Equations to Graph:
- \( y = \frac{1}{3}x + 5 \)
- \( y = -2x - 2 \)
Step 1: Create a Coordinate System
- Draw the Axes: Start by drawing the x-axis (horizontal) and y-axis (vertical) on a piece of graph paper or a blank sheet.
- Label the Axes: Mark the origin (0, 0) where the two axes intersect. Label the x-axis with values (for example, -10, -5, 0, 5, 10) and the y-axis with values (for example, -10, -5, 0, 5, 10). Choose an appropriate scale that allows you to graph both equations comfortably.
Step 2: Graph the First Equation
Equation 1: \( y = \frac{1}{3}x + 5 \)
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Determine the y-intercept: For the equation \( y = \frac{1}{3}x + 5 \), the y-intercept is \( 5 \). Plot a point at (0, 5).
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Determine another point using the slope: The slope is \( \frac{1}{3} \), which means for every 1 unit you move up in y, you move 3 units to the right in x.
- From the point (0, 5), move 3 units to the right (to x = 3) and 1 unit up (to y = 6). Plot this second point at (3, 6).
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Draw the line: Use a ruler to connect the two points (0, 5) and (3, 6) with a straight line. Extend the line across the graph, adding arrows on both ends to indicate that it continues indefinitely.
Step 3: Graph the Second Equation
Equation 2: \( y = -2x - 2 \)
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Determine the y-intercept: For the equation \( y = -2x - 2 \), the y-intercept is -2. Plot a point at (0, -2).
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Determine another point using the slope: The slope is -2, which means for every 1 unit you move up in y, you move 1 unit to the left in x and 2 units down in y.
- From the point (0, -2), you can move 1 unit to the right (to x = 1) and down 2 units (to y = -4). Plot this second point at (1, -4).
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Draw the line: Use a ruler to connect the two points (0, -2) and (1, -4) with a straight line. Extend this line across the graph, adding arrows on both ends to indicate that it continues indefinitely.
Step 4: Identify the Intersection Point
- Look for the Point of Intersection: After both lines have been drawn, observe the graph to find where the two lines intersect.
- Estimate the Coordinates: The coordinates of the intersection point will be the solution to the system of equations (the values of x and y that satisfy both equations).
Step 5: Write Down the Solution
Once you determine where the lines intersect, note the coordinates of this point. This point gives you the solution to the system of equations.
Example of Intersection
- You might find the intersection point to be approximately (x, y) coordinates from your graph. Make sure to double-check if your graph is accurately plotted.
Conclusion
By following these steps, you will have successfully graphed the two equations and found their intersection point, which represents the solution to the system of equations. If you wish, you can also verify the solution algebraically by substituting the x-value back into both original equations to check if it yields the same y-value.
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