Solve the following system of equations graphically on the set of axes below.

Question

y, equals, x, minus, 7
y=x−7
y, equals, minus, one third, x, minus, 3
y=−
3
1
​
x−3
Plot two lines by clicking the graph.
Click a line to delete it.
Solution:
give detailed instructions on how to graph

Answer 1

To solve the system of equations graphically, we need to graph the equations on a set of axes. In this case, we have two equations to plot:

1. \( y = x - 7 \)
2. \( y = -\frac{1}{3}x - 3 \)

Here's a step-by-step guide on how to graph these lines manually (or using graphing software, if available):

Step-by-Step Instructions

1. Identify the equations:

   • Line 1: \( y = x - 7 \)
   • Line 2: \( y = -\frac{1}{3}x - 3 \)

2. Graphing Line 1: \( y = x - 7 \)

   • Find the y-intercept: Set \( x = 0 \) (to find where the line crosses the y-axis).

     • \( y = 0 - 7 = -7 \)
     • So, the y-intercept is (0, -7).

   • Find another point: Choose another value for \( x \), say \( x = 7 \).

     • \( y = 7 - 7 = 0 \)
     • So, another point is (7, 0).

   • Plot the points on the graph: Draw the points (0, -7) and (7, 0) on the graph.

   • Draw the line: Connect the two points with a straight line, extending it across the graph.

3. Graphing Line 2: \( y = -\frac{1}{3}x - 3 \)

   • Find the y-intercept: Again, set \( x = 0 \).

     • \( y = -\frac{1}{3}(0) - 3 = -3 \)
     • So, the y-intercept is (0, -3).

   • Find another point: Choose another value for \( x \), let’s say \( x = 3 \).

     • \( y = -\frac{1}{3}(3) - 3 = -1 - 3 = -4 \)
     • So, another point is (3, -4).

   • Plot the points on the graph: Draw the points (0, -3) and (3, -4) on the graph.

   • Draw the line: Connect the two points with a straight line, extending it across the graph.

4. Determine the intersection: The solution to the system of equations is the point where the two lines intersect.

5. Identify the intersection point: Look for the point where the two lines cross on the graph. You may also choose to solve for the intersection algebraically if needed.

Summary of the Intersection Point (Optional)

After plotting the lines, observe where they cross. You can also solve the system of equations simultaneously to confirm the intersection point:

To find it algebraically, set the right-hand sides equal:

\[ x - 7 = -\frac{1}{3}x - 3 \]

Combine and solve for x:

1. Get rid of the fractions by multiplying both sides by 3:

   \[ 3(x - 7) = -x - 9 \]

2. Distribute:

   \[ 3x - 21 = -x - 9 \]

3. Add \( x \) to both sides:

   \[ 3x + x - 21 = -9 \]

4. Combine like terms:

   \[ 4x - 21 = -9 \]

5. Add 21 to both sides:

   \[ 4x = 12 \]

6. Divide by 4:

   \[ x = 3 \]

Now substitute \( x = 3 \) into one of the original equations to find \( y \):

\[ y = 3 - 7 = -4 \]

So, the intersection point is \( (3, -4) \).

Conclusion

By following these instructions, you can graph the two lines and find their intersection point both graphically and algebraically. The solution to the system is \( (3, -4) \).