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Chapter 1: Problem 9

Graph. Find the slope and the \(y\) -intercept. $$ y=-2 x+3 $$

Short Answer

Expert verified
Slope: \(-2\), y-intercept: \(3\)

Step by step solution

01

Identify the Slope-Intercept Form

The given equation is in the form: \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.
02

Determine the Slope

Compare the given equation \(y = -2x + 3\) with the slope-intercept form. The slope \(m\) is the coefficient of \(x\), which is \(-2\).
03

Find the y-intercept

In the given equation, the y-intercept \(b\) is the constant term, which is \(3\).
04

Interpret the Results

The slope of the line is \(-2\) and the y-intercept is \(3\). This means the line crosses the y-axis at \(3\) and for every unit increase in \(x\), \(y\) decreases by \(-2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form is an equation of a line in the format: \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) represents the y-intercept. This form is highly useful for quickly identifying the slope and the y-intercept of a line.
For example, in the equation \( y = -2x + 3 \), we can see that the coefficient of \( x \) (\( -2 \)) is the slope, and the constant term (\( 3 \)) is the y-intercept. This clear structure allows us to easily draw the graph of the line by simply knowing \( m \) and \( b \).
Slope
The slope (\( m \)) of a line measures its steepness. It tells us how much \( y \) changes for a unit change in \( x \). A positive slope means the line goes upwards as we move from left to right, while a negative slope means it goes downwards.
Using our example, the slope \( m \) is \( -2 \). This means that for every increase by 1 unit in \( x \), \( y \) decreases by \( 2 \) units.
In other words, if you move 1 step to the right on the x-axis, you move 2 steps downward on the y-axis.
Y-Intercept
The y-intercept (\( b \)) is the point where the line crosses the y-axis. This occurs when \( x = 0 \).
In our equation \( y =-2x+3 \), the y-intercept \( b \) is \( 3 \). This tells us that the point (0, 3) is on the line.
This is a crucial piece of information for graphing since it gives an exact starting point. From here, you can use the slope to determine the direction and steepness of the line.
Graphing Linear Equations
Graphing a linear equation involves just a few simple steps.
  • Step 1: Identify the slope-intercept form (\( y = mx + b \)).
  • Step 2: Find the slope (\( m \)) and the y-intercept (\( b \)).

In our example, \( y = -2x + 3 \), we already determined that the slope is \( -2 \) and the y-intercept is \( 3 \).
To graph this:
  • Start at the y-intercept (0, 3).
  • From this point, use the slope: go 1 unit to the right (positive direction on the x-axis) and 2 units down (negative direction on the y-axis).
  • Mark this new point.

You can repeat this move to plot more points.
Finally, draw a line through these points to graph your linear equation.

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