Problem 94 - Precalculus Essentials

Question

Describe how to write the equation of a line if the coordinates of two points along the line are known.

Step-by-Step Solution

Answer

To write the equation of a line, first calculate the slope using the formula \(m = \frac{y2 - y1}{x2 - x1}\). Then substitute the slope into the equation \(y = mx + b\). Using one of the given points, substitute the x and y values into the equation \(y = mx + b\) and solve for \(b\). The final equation of the line is thus obtained.

Step 1: Calculate the slope

Given two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) of the line passing through these points can be calculated using the formula: \(m = \frac{y2 - y1}{x2 - x1}\). Substitute the given points into the formula to find the slope.

Step 2: Use slope to form a partial equation

After obtaining the slope, the next step is to substitute this into the equation \(y = mx + b\). At this stage, the equation will be in the form \(y = mx + b\), where \(m\) is the known slope.

Step 3: Solve for the y-intercept

To find \(b\), we will use one of the given points. Substitute the x and y values of this point into the equation \(y = mx + b\). From there, rearrange the equation to solve for \(b\).

Step 4: Write the final equation of the line

Having found both the slope and the y-intercept, the final equation can be written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This is the equation of the line passing through the two given points.