Equations of Straight Lines (y = mx + c)

The gradient-intercept form of a linear equation is expressed as $y = mx + c$, where $x$ and $y$ are variables representing coordinates on the plane.

The parameter $m$ represents the gradient (or slope), which quantifies the rate of change of $y$ with respect to $x$.

The parameter $c$ represents the y-intercept, indicating the specific $y$-coordinate where the line intersects the vertical axis (at the point $(0, c)$).

This form is highly useful because it allows for the immediate visualization of the line's steepness and its starting position on the y-axis.

The gradient measures the 'steepness' of the line and is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points.

Mathematically, for two points $(x_1, y_1)$ and $(x_2, y_2)$, the gradient is given by the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

A positive gradient ($m > 0$) indicates the line slopes upwards from left to right, while a negative gradient ($m < 0$) indicates a downward slope.

A gradient of zero ($m = 0$) results in a horizontal line, whereas an undefined gradient (infinite steepness) represents a vertical line.

From a Graph: Identify the y-intercept ($c$) by looking at the y-axis intersection, then calculate the gradient ($m$) by selecting two clear points and finding $\frac{\text{rise}}{\text{run}}$.

Given Gradient and One Point: Substitute the known $m$ and the coordinates $(x, y)$ into $y = mx + c$ to solve for the unknown constant $c$.

Given Two Points: First, calculate the gradient $m$ using the two-point formula. Then, use either point to solve for $c$ as described in the previous method.

Rearranging Equations: If an equation is given in general form (e.g., $ax + by + d = 0$), it must be rearranged to isolate $y$ to correctly identify the gradient and intercept.

Parallel lines share the exact same steepness but have different y-intercepts; if they had the same intercept, they would be the same line.

Perpendicular lines have gradients that are negative reciprocals of each other (e.g., if $m_1 = 2$, then $m_2 = -\frac{1}{2}$).

Always isolate y: Before stating the gradient, ensure the equation is exactly in the form $y = mx + c$. For example, in $2y = 6x + 4$, the gradient is $3$, not $6$.

Check the sign: A common mistake is losing a negative sign during rearrangement or gradient calculation. Verify if the line on the graph matches the sign of your calculated $m$.

Sanity Check: If the line goes through the origin, $c$ must be $0$. If the line is 'downhill', $m$ must be negative.

Substitution Verification: Once you find the final equation, substitute the coordinates of a known point back into it to ensure the left side equals the right side.