How to find the gradient of a line?
Understand the Problem
The question is asking how to calculate the gradient (slope) of a line, which involves understanding the formula used for finding the gradient given two points or for a linear equation.
Answer
The gradient of the line is $m = 1$.
Answer for screen readers
The gradient of the line is $m = 1$.
Steps to Solve
- Identify the formula for gradient The formula for calculating the gradient (slope) $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
- Substitute the values of the points Plug in the coordinates of the points into the formula. For example, if the points are $(1, 2)$ and $(3, 4)$, then:
$$ m = \frac{4 - 2}{3 - 1} $$
- Perform the calculations Calculate the differences in the numerator and the denominator, and then divide them:
- Numerator: $4 - 2 = 2$
- Denominator: $3 - 1 = 2$
Putting it all together:
$$ m = \frac{2}{2} $$
- Simplify the result Now simplify the fraction:
$$ m = 1 $$
This is the gradient of the line.
The gradient of the line is $m = 1$.
More Information
The gradient represents how steep the line is. A gradient of 1 means that for every unit you move horizontally, you also move one unit vertically, indicating a 45-degree angle with the horizontal axis.
Tips
- Forgetting to subtract $y_1$ from $y_2$ and $x_1$ from $x_2$ in the correct order, which can lead to incorrect results.
- Not simplifying the fraction after calculating the differences.
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