Question
Plot the points(−4,5) and (8,9) in a rectangular coordinate system and find and interpret the slope of the line.
Answers
GPT-4o mini
To plot the points \((-4, 5)\) and \( (8, 9)\) in a rectangular coordinate system (also known as a Cartesian coordinate system), follow these steps:
1. **Plot the points**:
- The point \((-4, 5)\) is located 4 units to the left of the origin (on the x-axis) and 5 units up (on the y-axis).
- The point \((8, 9)\) is located 8 units to the right of the origin (on the x-axis) and 9 units up (on the y-axis).
You can label these points on a graph at their respective coordinates.
2. **Find the slope of the line**:
The slope \(m\) of the line passing through the points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting in the coordinates of the two points:
\[
(x_1, y_1) = (-4, 5) \quad \text{and} \quad (x_2, y_2) = (8, 9)
\]
We get:
\[
m = \frac{9 - 5}{8 - (-4)} = \frac{4}{8 + 4} = \frac{4}{12} = \frac{1}{3}
\]
3. **Interpret the slope**:
The slope \(m = \frac{1}{3}\) means that for every 1 unit increase in the \(y\)-direction (vertical), there is a corresponding increase of 3 units in the \(x\)-direction (horizontal). This indicates a gentle upward slope from left to right. In practical terms, if you were to move along the line that passes through these points, for every 3 units you move to the right, you would move up 1 unit.
This can be visually confirmed by drawing the line connecting the two points on the coordinate plane, where the line’s steepness would clearly reflect the calculated slope.
1. **Plot the points**:
- The point \((-4, 5)\) is located 4 units to the left of the origin (on the x-axis) and 5 units up (on the y-axis).
- The point \((8, 9)\) is located 8 units to the right of the origin (on the x-axis) and 9 units up (on the y-axis).
You can label these points on a graph at their respective coordinates.
2. **Find the slope of the line**:
The slope \(m\) of the line passing through the points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting in the coordinates of the two points:
\[
(x_1, y_1) = (-4, 5) \quad \text{and} \quad (x_2, y_2) = (8, 9)
\]
We get:
\[
m = \frac{9 - 5}{8 - (-4)} = \frac{4}{8 + 4} = \frac{4}{12} = \frac{1}{3}
\]
3. **Interpret the slope**:
The slope \(m = \frac{1}{3}\) means that for every 1 unit increase in the \(y\)-direction (vertical), there is a corresponding increase of 3 units in the \(x\)-direction (horizontal). This indicates a gentle upward slope from left to right. In practical terms, if you were to move along the line that passes through these points, for every 3 units you move to the right, you would move up 1 unit.
This can be visually confirmed by drawing the line connecting the two points on the coordinate plane, where the line’s steepness would clearly reflect the calculated slope.