Question

Graph these equations:
x+y=3
x+5y=

5
Click to select points on the graph.

Answers

GPT-4o mini
To graph the equations, we will first rewrite them in slope-intercept form (y = mx + b) for easier plotting.

1. **Equation: x + y = 3**
- Rearranging gives us:
\(y = -x + 3\)

2. **Equation: x + 5y = -5**
- Rearranging gives us:
\(5y = -x - 5\)
\(y = -\frac{1}{5}x - 1\)

Now, we can find points for each line to help us with the graphing.

### For the equation y = -x + 3:
- When \(x = 0\):
\(y = 3\) → Point (0, 3)
- When \(x = 3\):
\(y = 0\) → Point (3, 0)

### For the equation y = -\frac{1}{5}x - 1:
- When \(x = 0\):
\(y = -1\) → Point (0, -1)
- When \(x = 5\):
\(y = -2\) → Point (5, -2)

### Now, let's plot the points:

1. For **y = -x + 3**, plot the points (0, 3) and (3, 0) and draw the line through them.
2. For **y = -\frac{1}{5}x - 1**, plot the points (0, -1) and (5, -2) and draw the line through these points.

### Graphing
While I can't physically plot the graph here, you would plot the points on graph paper or using a software tool to see the two lines intersect.

- The first line will slope downwards from left to right and crosses the y-axis at (0, 3) and the x-axis at (3, 0).
- The second line also slopes downwards but is less steep, crossing the y-axis at (0, -1) and gradually decreases toward the right.

To find the intersection point of the two lines, you can solve the equations simultaneously:

1. Substitute \(y\) from the first equation into the second equation:
- \(x + 5(-x + 3) = -5\)
- \(x - 5x + 15 = -5\)
- \(-4x + 15 = -5\)
- \(-4x = -20\)
- \(x = 5\)

2. Now find \(y\):
\(y = -5 + 3 = -2\)

So the point of intersection is (5, -2).

You can now visualize the lines intersecting at this point! If you want to see it plotted visually, you can use any graphing tool or software like Desmos or GeoGebra.