Gradient Of A Line - GCSE Maths - Steps, Examples & Worksheet - Free Printable

Problem Description:

The image explains the concept of the gradient (slope) of a straight line and provides a formula to calculate it. The gradient is defined as:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

This formula represents the change in the \( y \)-coordinates divided by the change in the \( x \)-coordinates between two points on the line. The example given uses the points \((2, 1)\) and \((4, 5)\) to calculate the gradient.

Task:

We are asked to solve the problem using the provided formula and explain the solution step by step.

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Solution:

#### Step 1: Identify the coordinates of the points
From the image, the two points on the line are:
- Point 1: \((x_1, y_1) = (2, 1)\)
- Point 2: \((x_2, y_2) = (4, 5)\)

#### Step 2: Recall the formula for the gradient
The formula for the gradient \( m \) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

#### Step 3: Substitute the coordinates into the formula
Substitute \( (x_1, y_1) = (2, 1) \) and \( (x_2, y_2) = (4, 5) \) into the formula:

\[
m = \frac{5 - 1}{4 - 2}
\]

#### Step 4: Simplify the expression
Calculate the numerator and the denominator separately:
- Numerator: \( 5 - 1 = 4 \)
- Denominator: \( 4 - 2 = 2 \)

So, the expression becomes:

\[
m = \frac{4}{2}
\]

#### Step 5: Perform the division
Simplify the fraction:

\[
m = 2
\]

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Final Answer:

The gradient of the line passing through the points \((2, 1)\) and \((4, 5)\) is:

\[
\boxed{2}
\]

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Explanation:

1. Concept of Gradient: The gradient measures the steepness of a line. A positive gradient indicates an upward slope, while a negative gradient indicates a downward slope.
2. Formula Derivation: The formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) calculates the ratio of the vertical change (\( y_2 - y_1 \)) to the horizontal change (\( x_2 - x_1 \)). This is often referred to as "rise over run."
3. Application: By substituting the given coordinates into the formula, we can compute the gradient systematically.
4. Interpretation: A gradient of 2 means that for every unit increase in \( x \), \( y \) increases by 2 units. This matches the visual representation in the graph, where moving from \( x = 2 \) to \( x = 4 \) results in a rise from \( y = 1 \) to \( y = 5 \).

Thus, the solution is consistent with both the mathematical calculation and the graphical representation.