Question
Graph these equations:
x+y=3
x+5y=
–
5
Click to select points on the graph.
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.
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.