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Chapter 3: Problem 41

Graph each linear equation. \(y=-\frac{3}{4} x+3\)

Short Answer

Expert verified
Plot the y-intercept (0, 3), use the slope to plot (4, 0), then draw the line through the points.

Step by step solution

01

Identify the Slope and Y-Intercept

The given equation is in the slope-intercept form, which is written as \(y = mx + b\). Here, \(m\) is the slope and \(b\) is the y-intercept. For the equation \(y = -\frac{3}{4}x + 3\), the slope \(m\) is \(-\frac{3}{4}\) and the y-intercept \(b\) is 3.
02

Plot the Y-Intercept

Begin by plotting the y-intercept on the graph. Since the y-intercept is 3, place a point at (0, 3) on the y-axis.
03

Use the Slope to Find Another Point

The slope \(-\frac{3}{4}\) means that for every 4 units you move to the right on the x-axis, you move 3 units down on the y-axis (because the slope is negative). Starting from the point (0, 3), move 4 units to the right (to x = 4) and 3 units down (to y = 0). Place a point at (4, 0).
04

Draw the Line

Draw a straight line through the two points (0, 3) and (4, 0) to graph the linear equation.

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Key Concepts

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

slope-intercept form
When dealing with linear equations, the slope-intercept form is one of the most common ways to express the equation of a line. This form is written as:
\( y = mx + b \)
Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis.
Understanding this form is crucial because it directly shows how the line behaves. The slope \( m \) tells us how steep the line is, and the y-intercept \( b \) provides a starting point for graphing. For the equation given, \( y = -\frac{3}{4} x + 3 \), the slope \( m \) is \( -\frac{3}{4} \) and the y-intercept \( b \) is 3.
plotting points
Once you have identified the slope and y-intercept, the next step is to plot these points on the graph. Let's start with the y-intercept.
  • Identify the y-intercept, which in our example is 3.
  • Plot this point on the y-axis, at (0, 3).

The next step is to use the slope to find another point. A slope of \( -\frac{3}{4} \) indicates that for every 4 units we move to the right along the x-axis, we will move 3 units down along the y-axis (the negative sign tells us to move down instead of up).
  • Starting from the y-intercept (0, 3), move 4 units to the right to (4, 3).
  • Then, move 3 units down to (4, 0).
Now, you have two points: (0, 3) and (4, 0).
y-intercept
The y-intercept is a critical point in graphing a linear equation. It is where the line crosses the y-axis. In our equation \( y = -\frac{3}{4} x + 3 \), the y-intercept is 3.
  • The y-intercept directly connects to the b-value in the slope-intercept form, \( y = mx + b \).
  • This tells us that regardless of the slope, every line begins its journey on the y-axis at point (0, b).
  • In our case, plot the point (0, 3) on the y-axis.
This point is your foundation when graphing your line.
slope
The slope of a line is a measure of its steepness. It tells us how much the y-value of a point on the line changes as the x-value changes.
In the equation \( y = -\frac{3}{4}x + 3 \), the slope \( m \) is \( -\frac{3}{4} \) which means:
  • For every 4 units you move to the right along the x-axis, you move 3 units down along the y-axis.

Here's how to interpret the slope:
  • A positive slope means the line rises from left to right.
  • A negative slope means the line falls from left to right.
  • A larger absolute value of the slope indicates a steeper line.

Knowing how to use the slope makes plotting a second point easy. From the y-intercept at (0, 3), move right 4 units and down 3 units to arrive at (4, 0). This confirms our points (0, 3) and (4, 0) on the graph.

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Most popular questions from this chapter

Find the \(x\) -intercept and the \(y\) -intercept for the graph of each equation. $$ x-4=0 $$

Solve each equation. $$ 9-x=-4 $$

For each pair of equations, give the slopes of the lines and then determine whether the two lines are parallel, perpendicular, or neither. See Example \(6 .\) $$ \begin{aligned} &2 x+5 y=4\\\ &4 x+10 y=1 \end{aligned} $$

Solve each inequality, and graph the solution set on a number line. Section \(2.8 .\) $$ 5-3 x \leq-10 $$

Solve each equation. $$ -5+t=3 $$
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