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The Anatomy of a Straight Line

When you walk up a steady hill, the steepness remains exactly the same with every step you take. In coordinate geometry, this constant steepness is the defining feature of a linear equation.

The standard way to represent these straight lines in OCR exams is the equation:

$y = mx + c$

In this formula, $m$ represents the gradient (how steep the line is), and $c$ represents the y-intercept (the exact point where the line crosses the vertical -axis). You can also think of the -intercept as the value of whenever the -coordinate is exactly .

Method 1: Equation from the Gradient and One Point

If you already know the gradient and one point on the line, you can find the full equation using substitution. You simply replace $m$ with the gradient, and $x$ and $y$ with the coordinates of your given point, which leaves only $c$ to be calculated.

Find the equation of the line with a gradient of $4$ that passes through the point $(2, 11)$ .

Step 1: Write the general equation and substitute the known values ( $m = 4$ , $x = 2$ , $y = 11$ ).

$y = mx + c$
$11 = 4(2) + c$

Step 2: Multiply the gradient by the $x$ -coordinate.

$11 = 8 + c$

Step 3: Rearrange the equation to solve for $c$ .

$11 - 8 = c$
$c = 3$

Step 4: Write the final equation by plugging $m$ and $c$ back into the general formula.

$y = 4x + 3$

Method 2: Equation from Two Points

Often, you will only be given two points on a line. Before you can find the $y$ -intercept, you must first calculate the gradient using the rise (change in vertical distance) divided by the run (change in horizontal distance).

The formula to find the gradient between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Once you have calculated $m$ , you can choose either of the two original points and repeat the substitution method from Method 1 to find $c$ .

Find the equation of the line passing through the points $A(-1, -5)$ and $B(3, 7)$ .

Step 1: Label your coordinates and calculate the gradient ( $m$ ).

$(x_1 = -1, y_1 = -5)$ and $(x_{}$

Step 2: Choose one point (e.g., $B(3, 7)$ ) and substitute $m, x,$ and $y$ into $y = mx + c$ .

$7 = 3(3) + c$

Step 3: Solve for the $y$ -intercept ( $c$ ).

$7 = 9 + c$
$c = 7 - 9 = -2$

Step 4: State the final equation.

$y = 3x - 2$

Special Cases: Horizontal and Vertical Lines

Not all lines fit perfectly into the standard format. A horizontal line has a gradient of $0$ , so its equation simplifies down to $y = c$ .

Conversely, a vertical line does not cross the $y$ -axis (unless it is the axis itself) and has an undefined gradient. Because its $x$ -value never changes, a vertical line cannot be written in $y = mx + c$ form and is simply written as $x = k$ (where is the -intercept).

Students frequently lose the final accuracy mark in OCR exams by simply writing the expression (e.g., $3x - 2$ ) instead of stating the full equation ( $y = 3x - 2$ ).

When using the gradient formula with negative coordinates, always wrap the negative numbers in brackets (e.g., $7 - (-5)$ ) to avoid the classic error of dropping a negative sign.

The command word "Calculate" means you must clearly show your substitution into $y = mx + c$ to secure method marks, even if you can spot the y-intercept mentally.

If an OCR question specifically asks for the equation in the form $ax + by + c = 0$ , you must rearrange your final $y = mx + c$ equation to make it equal zero and ensure all numbers are integers (no fractions).

The study of geometry using a coordinate system, allowing shapes and lines to be described algebraically.

An algebraic representation of a straight line where variables ( $x$ and $y$ ) are raised to a power no higher than 1.

A measure of the steepness of a line, calculated as the change in the $y$ -coordinate for every unit increase in the $x$ -coordinate.

The $y$ -coordinate of the exact point where a line intersects the vertical $y$ -axis, occurring when $x = 0$ .

The vertical change or difference between two points on a graph.

The horizontal change or difference between two points on a graph.