How do you find the gradient of a line?
Understand the Problem
The question is asking how to determine the gradient (slope) of a line, which is a fundamental concept in coordinate geometry. The gradient indicates how steep the line is and can be found using the change in y over the change in x (Δy/Δx).
Answer
The gradient (slope) of the line is calculated using $m = \frac{y_2 - y_1}{x_2 - x_1}$.
Answer for screen readers
The gradient (slope) of the line is given by the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Steps to Solve
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Identify the coordinates of two points Choose two points on the line for which you want to find the gradient. For example, let's say the points are ( A(x_1, y_1) ) and ( B(x_2, y_2) ).
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Calculate the change in y (Δy) Find the difference in the y-coordinates of the two points. This is done by using the formula: $$ \Delta y = y_2 - y_1 $$
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Calculate the change in x (Δx) Find the difference in the x-coordinates of the two points with the formula: $$ \Delta x = x_2 - x_1 $$
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Compute the gradient (slope) Now, divide the change in y by the change in x to get the gradient (slope) of the line: $$ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} $$
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Interpret the gradient If ( m > 0 ), the line slopes upwards from left to right; if ( m < 0 ), it slopes downwards. If ( m = 0 ), the line is horizontal.
The gradient (slope) of the line is given by the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
More Information
The gradient is a crucial concept in calculus and helps in understanding how functions behave. A steeper gradient indicates a more rapid change in y with respect to x.
Tips
- Confusing the order of coordinates when calculating ( \Delta y ) and ( \Delta x ).
- Forgetting to subtract the smaller value from the larger value in the changes, which can lead to a negative slope where incorrect.
- Not correctly identifying two distinct points on the line.
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