Problem 63 - Introductory Algebra for College Students

Question

Write an equation in the form \(y=m x+b\) of the line that is described. The line rises from left to right. It passes through the origin and a second point with equal \(x\) - and \(y\) -coordinates.

Step-by-Step Solution

Answer

The equation of the line is \(y=x\).

Step 1: Identifying Key Information

First, identify key pieces of information from the problem. It's known that the line passes through the origin (0,0) and a second point where x=y, and it rises from left to right.

Step 2: Determining the Slope and Intercepts

Since the line rises from the left to right and passes through the origin and a point where x=y, it can be concluded that the slope m =1. This is because the rise and run are the same, as x=y. And, because the line passes through the origin, the y-intercept b = 0.

Step 3: Writing the Line Equation

Use the slope m and y-intercept b found in step 2 to write the equation of the line. The equation of the line in slope-intercept form, \(y=mx+b\), becomes \(y=x+0\), simplifying to \(y=x\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most common ways to write the equation of a line. It provides a clear understanding of both the slope and the y-intercept of the line, which are essential for graphing and analyzing straight lines. The general formula is given by: \[y = mx + b\] where:

• \(m\) is the slope of the line
• \(b\) is the y-intercept, the point where the line crosses the y-axis

The slope-intercept form is a straightforward way to convey the steepness and position of a line on the coordinate plane. By converting a line equation into this form, you can easily graph it and understand its behavior, such as how quickly it rises or falls as you move along the x-axis.

Slope of a Line

The slope of a line is a measure of its steepness. It tells us how much the line goes up (or down) for every step it goes sideways. Mathematically, slope is the ratio of the rise over the run between two points on the line. If a line passes through two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) can be calculated as: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]In the context of the exercise, since the line passes through the origin \((0,0)\) and a second point where \(x = y\) , the slope \(m\) is actually \(1\). This is because for every increase in \(x\), \(y\) increases by the same amount. A positive slope, like in this case, indicates the line rises from left to right. If the slope were negative, the line would fall.

Equation of a Line

An equation of a line expresses a relationship between \(x\) and \(y\) coordinates of every point on the line. Once you know the slope \(m\) and the y-intercept \(b\), you can write the line's equation. From the exercise, we learned:

• The line passes through the origin \((0,0)\), making \(b = 0\).
• The slope \(m = 1\), implying equal increases in \(x\) and \(y\).

Putting these values in the slope-intercept formula \(y = mx + b\), we get: \[y = x +0\] or simply \[y = x\]. This equation means for every step in \(x\), \(y\) moves by the same step, perfectly depicting a 45-degree angle when graphed. The equation \(y=x\) signifies a direct one-to-one relationship between \(x\) and \(y\), with no shifts up or down, since there is no y-intercept other than \(0\).