Problem 72 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The equation of a line passing through two points can be written by first calculating the slope of the line using the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), substituting the slope into the equation \(y - y_1 = m(x - x_1)\) with the coordinates of either point, and then simplifying the equation to put it in slope-intercept form \(y = mx + b\).

Step 1: Understand the Problem

We are given two points on a line, for example, let's call them point A \((x_1, y_1)\) and point B \((x_2, y_2)\). We need to find the equation of the line that passes through these points.

Step 2: Calculate the Slope

The slope \(m\) of a line can be found using the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Plug the coordinates of the two given points into this formula to find the slope.

Step 3: Write the Equation of the Line

Next, the equation of the line that passes through a given point \((x_1, y_1)\) with a slope \(m\) can be written in slope-intercept form as \(y - y_1 = m(x - x_1)\). Substitute the calculated slope and the coordinates of either point into this formula to write the equation of the line.

Step 4: Simplify the Equation

The final step involves simplifying the equation from the previous step to put it into slope-intercept form, which is \(y = mx + b\). Expand the right side of the equation and then isolate \(y\) by adding \(y_1\) to both sides of the equation.

Key Concepts

Slope

The slope of a line is a measure that tells us how steep the line is. Imagine the line as a ramp; the steeper the ramp, the greater the slope. Mathematically, the slope is represented by the letter \( m \). To find the slope between two given points on a line, say point A \((x_1, y_1)\) and point B \((x_2, y_2)\), we use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

• The numerator \(y_2 - y_1\) represents the change in the \(y\)-values, or the vertical change.
• The denominator \(x_2 - x_1\) represents the change in the \(x\)-values, or the horizontal change.

This formula effectively tells us how many units the line moves up or down for every unit it moves to the right.If \( m \) is positive, the line slopes upwards. If \( m \) is negative, it slopes downwards. A zero slope means the line is horizontal, while an undefined slope (when \(x_2 = x_1\)) results in a vertical line.

Slope-Intercept Form

The slope-intercept form of a line's equation is one of the most popular ways to express the equation of a straight line. This form is particularly useful because it directly shows us the slope of the line and where the line intercepts the y-axis. The slope-intercept form is expressed as:\[y = mx + b\]

• \( m \) is the slope of the line, which we've already calculated using the two points.
• \( b \), known as the y-intercept, is the point where the line crosses the y-axis (where \( x = 0 \)).

To write the equation of a line in slope-intercept form: - First, use the formula \(y - y_1 = m(x - x_1)\), plugging in the slope \(m\) and the coordinates of one of the points. - Simplify the equation by expanding it and solving for \(y\) to isolate it on one side. - As a final step, find \(b\) by ensuring the equation is fully simplified to match \(y = mx + b\).This form is excellent for quickly sketching a line and understanding how it changes across a graph.

Points on a Line

When working with lines, points are critical because just two points can completely determine a straight line. Given two points on a line, often labeled as \((x_1, y_1)\) and \((x_2, y_2)\), we can derive important information about the line they form, including its slope and equation. Let's break down how we use these points:

• **Finding the Slope**: As mentioned, the slope \( m \) gives us a way to quantify the line's steepness, derived directly from these points.
• **Equation of the Line**: With the slope in hand, either of these two points can be used in the point-slope form, \(y - y_1 = m(x - x_1)\), to begin forming the full equation of the line.
• **Verification**: Once you have your line equation, you can check if other points lie on the same line by substituting their coordinates into the equation. If it balances (both sides equal), the point lies on the line.

Remember, a line extends infinitely in both directions, but we usually only need two points to lay down its path in the coordinate plane. These fundamental concepts work together to help us understand and manipulate linear equations efficiently.