Equations of Straight Lines (y = mx + c)

The slope-intercept form of a linear equation is expressed as $y = mx + c$, where every point $(x, y)$ on the line satisfies this equality.

The variable $m$ represents the gradient (or slope), which measures the steepness of the line and its direction (uphill or downhill).

The constant $c$ represents the y-intercept, which is the specific y-coordinate where the line crosses the y-axis (where $x = 0$).

Linear equations represent a constant rate of change; for every unit increase in $x$, the value of $y$ changes by exactly $m$ units.

The gradient is calculated as the ratio of the vertical change to the horizontal change between any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line.

Mathematically, the formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$, often referred to as $\frac{\text{rise}}{\text{run}}$.

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 perfectly horizontal line, as there is no vertical change regardless of the horizontal movement.

The y-intercept is the point where the line intersects the vertical axis; at this point, the x-coordinate is always zero.

In the equation $y = mx + c$, the value of $c$ is the y-value when $x=0$. For example, in $y = 2x + 5$, the line crosses the y-axis at $(0, 5)$.

If a line passes through the origin $(0, 0)$, the value of $c$ is zero, and the equation simplifies to $y = mx$.

The y-intercept provides a fixed reference point that allows for the easy plotting of the line's starting position on a graph.

Horizontal Lines: These have a gradient of $0$. Their equations take the form $y = k$, where $k$ is a constant representing the y-intercept.

Vertical Lines: These have an undefined gradient because the 'run' is zero. Their equations take the form $x = k$, where $k$ is the x-intercept.

Parallel Lines: Two lines are parallel if they have the exact same gradient ($m_1 = m_2$). They will never intersect because they rise and run at the same rate.

Perpendicular Lines: Two lines are perpendicular if they meet at a right angle ($90^{\circ}$). Their gradients are negative reciprocals of each other, meaning $m_1 \cdot m_2 = -1$.

From a Graph

• Identify the y-intercept ($c$) by looking at where the line crosses the vertical axis.
• Calculate the gradient ($m$) by selecting two clear points and finding the $\frac{\text{rise}}{\text{run}}$.
• Substitute $m$ and $c$ into the general form $y = mx + c$.

From a Point and Gradient

• Start with the general form $y = mx + c$ and substitute the known gradient $m$.
• Substitute the coordinates of the given point $(x, y)$ into the equation to solve for the unknown constant $c$.
• Rewrite the final equation with the calculated values for $m$ and $c$.

It is vital to distinguish between the gradient (the rate of change) and the intercept (the starting value). A high gradient means a steep line, while a high y-intercept simply means the line is shifted higher up the graph.

• Check the Sign: Always verify if your gradient's sign matches the graph. If the line goes 'downhill', your calculated $m$ must be negative.

• Rearrange First: If an equation is given in a different form (e.g., $2y - 4x = 10$), always rearrange it into $y = mx + c$ before identifying the gradient or intercept.

• Substitution Check: Once you have found your equation, substitute the coordinates of a known point back into it. If the left side equals the right side, your equation is correct.

• Avoid the 'Run/Rise' Trap: A common mistake is dividing the change in $x$ by the change in $y$. Always remember that $y$ (the vertical) goes on top: $\frac{\Delta y}{\Delta x}$.