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Chapter 2: Problem 48

For each line, ( \(a\) ) find the slope and ( \(b\) ) sketch the graph. $$-4 y=5 x$$

Short Answer

Expert verified
The slope is \( -\frac{5}{4} \). Sketch the graph by drawing a line through the origin with this slope.

Step by step solution

01

Rewrite the equation

First, rewrite the equation in slope-intercept form, which is in the form of \[ y = mx + b \], where \( m \) is the slope and \( b \) is the y-intercept.The given equation is\[ -4y = 5x. \]Divide both sides by -4 to isolate y:\[ y = -\frac{5}{4}x. \]
02

Identify the slope

In the equation \( y = -\frac{5}{4}x \), the coefficient of \( x \) is the slope.Thus, the slope \( m \) is\[ m = -\frac{5}{4}. \]
03

Sketch the graph

The equation \( y = -\frac{5}{4}x \) represents a straight line passing through the origin (0,0) with a slope of \( -\frac{5}{4} \).To sketch the graph, start at the origin and use the slope to find another point on the line. Because the slope is \( -\frac{5}{4} \), you can go down 5 units and right 4 units from the origin to find a second point.Connect these two points with a straight line.

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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 of a linear equation is one of the most essential concepts when dealing with linear equations. It is written as \( y = mx + b \). Here, \(m\) represents the slope of the line, and \(b\) stands for the y-intercept, which is the point where the line crosses the y-axis.

To find the slope-intercept form, you need to isolate \(y\) on one side of the equation. For example, in the given problem, the equation -4y = 5x was transformed into \( y = -\frac{5}{4}x \).

This form makes it straightforward to identify the slope and y-intercept, both critical for graphing and understanding the behavior of linear equations.
linear equations
Linear equations represent relationships where the rate of change is constant. These equations always graph as a straight line.

The general form of a linear equation is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. When using the slope-intercept form, you rewrite it to isolate \( y \).

Using our solved example, \(-4y = 5x\) became \( y = -\frac{5}{4}x \). Here, \(-\frac{5}{4}\) is the slope showing how steep the line is, and because there is no \( + b \) term, the y-intercept is zero.

Understanding how to manipulate these forms allows you to graph the equation and understand its properties better.
graphing lines
Graphing lines starts with understanding the slope and y-intercept. The slope indicates how much the line rises or falls for each unit it moves along the x-axis. A positive slope means the line goes upward, whereas a negative slope means it goes downward.

In our exercise, the slope is \(-\frac{5}{4}\), which means that for every 4 units moved to the right, the line goes down 5 units. Start by plotting the y-intercept, which is the origin (0,0) in this case. Then, use the slope to find another point.

From the origin, move down 5 units and to the right 4 units to find the second point. Draw a straight line through these points to complete the graph.

This method helps visualize how linear equations function in a coordinate system and gives a clear visual representation of the solution.

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Most popular questions from this chapter

Solve each problem. To visualize the situation, use graph paper and a pair of compasses to carefully draw the graphs of the circles. The locations of three receiving stations and the distances to the epicenter of an earthquake are contained in the following three equations: \((x-2)^{2}+(y-1)^{2}=25\) \((x+2)^{2}+(y-2)^{2}=16,\) and \((x-1)^{2}+(y+2)^{2}=9 .\) Determine the location of the epicenter.

Show that \((f \circ g)(x)\) is not equivalent to \((g \circ f)(x)\) for $$f(x)=3 x-2 \quad \text { and } \quad g(x)=2 x-3$$

The tables give some selected ordered pairs for functions \(f\) and \(g\). $$\begin{array}{|c|c|c|c|}\hline x & 3 & 4 & 6 \\\\\hline f(x) & 1 & 3 & 9 \\\\\hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|}\hline x & 2 & 7 & 1 & 9 \\\\\hline g(x) & 3 & 6 & 9 & 12 \\\\\hline\end{array}$$ Find each of the following. $$(g \circ f)(3)$$

Give the slope and y-intercept of each line, and graph it. $$4 y=-3 x$$

Solve each problem. To visualize the situation, use graph paper and a pair of compasses to carefully draw the graphs of the circles. Suppose that receiving stations \(X, Y,\) and \(Z\) are located on a coordinate plane at the points \((7,4),(-9,-4),\) and \((-3,9),\) respectively. The epicenter of an earthquake is determined to be 5 units from \(X, 13\) units from \(Y\), and 10 units from \(Z\). Where on the coordinate plane is the epicenter located?
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