Writing Linear Equations

Writing the equation of a line is a core skill in algebra. Depending on the information you are given, you can build a linear equation using two primary formulas:

1. Slope-Intercept Form: $y = mx + b$
   • $m$ is the slope (rate of change).
   • $b$ is the $y$-intercept (starting value).
2. Point-Slope Form: $y - y_1 = m(x - x_1)$
   • $m$ is the slope.
   • $(x_1, y_1)$ is a known point on the line.

Given a Point and a Slope

When you know the slope of a line and exactly one point it passes through, start with the point-slope form and simplify it into slope-intercept form.

Example: Write the equation of the line with a slope of $-2$ passing through $(3, 1)$.

1. Identify your values: $m = -2$, $x_1 = 3$, $y_1 = 1$.
2. Plug them into the point-slope formula: $y - 1 = -2(x - 3)$
3. Distribute the slope: $y - 1 = -2x + 6$
4. Add $1$ to both sides to isolate $y$: $y = -2x + 7$

Given Two Points

If you only have two points, you first need to calculate the slope using the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$. Once you have the slope, use either point and the point-slope form to find the equation.

Example: Write the equation of the line passing through $(1, 3)$ and $(4, 9)$.

1. Find the slope ($m$): $m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2$
2. Choose one point, say $(1, 3)$, and use the point-slope form: $y - 3 = 2(x - 1)$
3. Distribute and solve for $y$: $y - 3 = 2x - 2$ $y = 2x + 1$

From a Word Description

In real-world scenarios, lines are often described by a starting amount and a rate of change.

• The rate of change is your slope ($m$). If something is decreasing, the slope is negative.
• The starting amount is your $y$-intercept ($b$).

Example: A pool starts with 200 liters of water and drains at a rate of 15 liters per hour. Write the equation for the amount of water ($y$) after $x$ hours.

1. The starting amount is 200 liters, so $b = 200$.
2. The pool is draining 15 liters per hour, so the rate of change is negative: $m = -15$.
3. Plug these directly into the slope-intercept form ($y = mx + b$): $y = -15x + 200$