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Chapter 2: Problem 50

Graph each equation. $$ \frac{4}{5}=-\frac{1}{3} x-\frac{3}{4} y $$

Short Answer

Expert verified
The graph of the equation is a straight line with a slope of (-4/9) and a y-intercept of (-16/15).

Step by step solution

01

Rewrite the equation in slope-intercept form

Start by solving the equation for y to get it into the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. Multiply each term by the least common denominator, which is 60, to eliminate the fractions: 48 = -20x - 45y.
02

Isolate the y-term

Add 20x to both sides to get all y-terms on one side and x-terms on the other: 48 + 20x = -45y.
03

Solve for y

Divide each term by -45 to solve for y: y = (-20/45)x - (48/45). Simplify the equation to get y = (-4/9)x - (16/15).
04

Identify the slope and y-intercept

From the equation y = (-4/9)x - (16/15), we can see that the slope m is -4/9 and the y-intercept b is -16/15.
05

Plot the y-intercept on the y-axis

Locate the point (0, -16/15) on the y-axis and place a dot, as this is the point where the line will cross the y-axis.
06

Use the slope to find a second point

From the y-intercept, use the slope to find another point on the line. Since the slope is -4/9, you can go down 4 units and to the right 9 units to find a new point on the graph.
07

Draw the line

Connect the y-intercept with the second point you found using the slope and draw the line through these two points.

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Key Concepts

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

Slope-intercept form
Understanding the slope-intercept form of a linear equation is crucial for graphing lines. This form is expressed as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis.

To convert a linear equation to slope-intercept form, as we did in the exercise with \( \frac{4}{5} = -\frac{1}{3}x - \frac{3}{4}y \), we isolate \( y \) to the left side. The slope-intercept form makes it immediately clear what the slope and y-intercept are, which are essential for graphing the line.
Solving for y
Solving an equation for \( y \) is a step to get to the slope-intercept form. To start, you might need to clear any fractions by multiplying each term by the least common denominator, as you will see with equations like \( \frac{4}{5} = -\frac{1}{3}x - \frac{3}{4}y \). After removing fractions, you'll rearrange the terms to collect the \( y \)-terms on one side and the \( x \)-terms on the other. Lastly, divide by the coefficient of \( y \) to isolate \( y \).

This process gives you the slope-intercept form, enabling you to graph the equation with ease because you'll have identified both the slope and y-intercept directly within the equation.
Plotting points on a graph
Plotting points on a graph is fundamental to visualizing and understanding the behavior of linear equations. Start with the y-intercept, which is where you'll place your first point on the graph. In our example, the y-intercept is \( -\frac{16}{15} \), corresponding to the coordinate (0, \( -\frac{16}{15} \)).

Next, use the slope to find your second point. If your slope is \( -\frac{4}{9} \), this means you鈥檒l move down 4 units and to the right 9 units from the y-intercept. These movements on the graph create a second point through which your line will pass. With two points plotted, you can easily draw the line representing your equation.

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Most popular questions from this chapter

Graph each absolute value inequality. $$ 1-y<|2 x-1| $$

a. Graph the equations \(y=\left|\frac{1}{2} x-6\right|+3\) and \(y=-\left|\frac{1}{2} x+6\right|-3\) on the same set of axes. b. Writing Describe the similarities and differences in the graphs.

Evaluate each function for five values of \(x .\) Then graph each function. \(f(x)=|2 x+1|-3\)

Find the slope of each line. $$ \frac{x}{2}+\frac{y}{3}=1 $$

Graph each absolute value equation. $$ y=-|-2 x-1|+1 $$
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