Problem 43 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ -y=-x+3 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is 1 and the y-intercept is -3.

Step 1: Rewrite the equation in slope-intercept form

To rewrite the equation in slope-intercept form, we need to isolate y on one side of the equation. Given the equation \(-y=-x+3\), we will multiply both sides by \(-1\) to make the coefficient of y positive: $$ y=x-3 $$

Step 2: Identify the slope and y-intercept

Now that the equation is in the form \(y=mx+b\), we can easily identify the slope and y-intercept. In our case, we have: $$y = x - 3$$ Comparing this with the general slope-intercept form, we can see that the slope \(m = 1\) and the y-intercept \(b = -3\). So, the slope of the line is 1 and the y-intercept is -3.

Key Concepts

Equation Rewriting

The first step when working with linear equations is often to rewrite them in a more useful form. The slope-intercept form is particularly helpful because it clearly shows the slope and the y-intercept of a line.
Given an equation like \(-y = -x + 3\), we aim to express it in the form \(y = mx + b\). Here, \(m\) represents the slope and \(b\) represents the y-intercept.
To do this effectively, you need to isolate \(y\) on one side. For example, multiply both sides by \(-1\) in this equation:

• The equation becomes \(y = x - 3\).

This transformation maintains the balance of the equation while allowing us to clearly recognize the slope and intercept.

Slope Determination

Once we have the equation in the form \(y = mx + b\), finding the slope becomes straightforward.
The slope \(m\) represents how much \(y\) changes for every unit increase in \(x\). In our equation \(y = x - 3\), it is compared with the standard slope-intercept form \(y = mx + b\):

• The coefficient of \(x\) is \(1\), so the slope \(m = 1\).

The slope of \(1\) indicates that for every additional step in \(x\), \(y\) increases by one unit. Understanding the slope can help you visualize how the line tilts on the graph.

Y-intercept Identification

After rewriting the equation in slope-intercept form, identifying the y-intercept is simple.
The y-intercept \(b\) is the value of \(y\) when \(x = 0\). In the equation \(y = x - 3\), \(b\) is directly given as \(-3\).

• This means the line crosses the y-axis at \(-3\).

Knowing the y-intercept is crucial when you begin plotting points of the line on a graph. It's your starting point on the y-axis and helps set the position of the line in relation to the graph's axes.