Problem 12

Question

Find the slope and the $y$ -intercept of the line with the given equation. $$y=5-x$$

Step-by-Step Solution

Verified

Answer

The slope of the line is '-1' and the y-intercept is '5'.

1Step 1: Identify the slope

The equation given is $y = 5 - x$. In this equation, the coefficient of $x$ is '-1'. Therefore, the slope of the line ($m$) is '-1'.

2Step 2: Identify the y-intercept

The y-intercept is the value of $y$ when $x = 0$. In our equation, it is the constant term. Hence, the y-intercept ($c$) is '5'.

Key Concepts

Understanding the SlopeGrasping the Y-interceptEquation of a Line

Understanding the Slope

The slope of a line is a measure of its steepness. It's represented by the letter "m" in the equation of a line in the form $ y = mx + c $. The slope describes how much the y-coordinate of a point on the line changes for a one-unit increase in the x-coordinate.
For example:

If the slope is positive, the line rises as you move from left to right.
If the slope is negative, the line falls as you go from left to right.
If the slope is zero, the line is horizontal.
If the slope is undefined, the line is vertical.

In the example equation $ y = 5 - x $, we can rewrite it in standard slope-intercept form as $ y = -1x + 5 $. Hence, the slope "m" is $-1$, meaning the line decreases by one unit vertically for every one unit it moves horizontally.

Grasping the Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. It indicates the value of $ y $ when $ x $ is zero. In the standard linear equation form $ y = mx + c $, the y-intercept is represented by the constant "c."
Understanding the y-intercept helps to determine where the line will intersect the vertical axis on a graph.

A positive y-intercept shifts the line upwards.
A negative y-intercept shifts the line downwards.

For the equation $ y = 5 - x $, setting $ x = 0 $ results in $ y = 5 $. This tells us that the line crosses the y-axis at $ y = 5 $. Thus, the y-intercept in this case is 5.

Equation of a Line

The equation of a line in two-dimensional space fundamentally relates $ x $ and $ y $. It can predict or describe points on the line. The most common forms of line equations include the slope-intercept form $ y = mx + c $, and the point-slope form $ y - y_1 = m(x - x_1) $.
In the slope-intercept form: