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Chapter 2: Problem 64

a. Rewrite the given equation in slope-intercept form. b. Give the slope and \(y\) -intercept. c. Use the slope and y-intercept to graph the linear function. \(6 x-5 y-20-0\)

Short Answer

Expert verified
The slope-intercept form of the given equation is \(y = 6/5x + 4\). The slope of the equation is 6/5 and the y-intercept is 4. By plotting the y-intercept and using the slope to find additional points, we can accurately graph the function.

Step by step solution

01

Rewrite the equation in slope-intercept form

The slope-intercept form of a linear equation is \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. So, first we rearrange the given equation \(6x - 5y - 20 = 0\) to this form. This is done by moving the terms containing \(y\) to the left side of the equation and the others to the right side. The equation, therefore, becomes \(5y = 6x + 20\). To isolate \(y\), we divide the entire equation by 5: So, \(y = 6/5x + 4\).
02

Determine the slope and y-intercept

From the rewritten equation, we can read the slope \(\(m\)\) and y-intercept \(\(b\)\). According to the form \(y=mx+b\), we see that the slope \(\(m\)\) is 6/5 and the y-intercept \(\(b\)\) is 4.
03

Graph the linear function

To graph the linear function, first we note the y-intercept (0, 4) which gives us the point where the line intersects the y-axis. Then, the slope 6/5 tells us that for each step of 1 to the right (positive x direction), we move 6/5 steps upwards (positive y direction), and thus get more points which the line goes through. Drawing a straight line through all these points gives the graph of the function.

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Key Concepts

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

Linear Functions
Linear functions are a fundamental concept in algebra. They describe a straight line on a graph. The general form of a linear equation is \( ax + by + c = 0 \), where \( a \), \( b \), and \( c \) are constants. However, it's often more useful to use the slope-intercept form \( y = mx + b \). Here, \( m \) represents the slope of the line and \( b \) is the y-intercept.
  • The slope \( m \) indicates the steepness and direction of the line. A positive slope means the line rises from left to right, while a negative slope indicates it falls.
  • The y-intercept \( b \) shows where the line crosses the y-axis. It provides a starting point for graphing the line.
Understanding linear functions allows you to explore relationships between variables that have constant rates of change. They are everywhere in real life, from calculating distance to interpreting financial data.
Graphing Equations
Graphing equations involves translating an algebraic expression into a visual form, which is a graph. For linear equations in the slope-intercept form \( y = mx + b \), graphing becomes straightforward:
  • Begin with the y-intercept \( (0, b) \). This is the point where the line crosses the y-axis.
  • Use the slope \( m \) to find additional points on the graph. If the slope is a fraction like \( \frac{6}{5} \), it means for every 5 units you go right on the x-axis, you move 6 units up on the y-axis.
  • Connect these points to form a straight line.
Graphing helps visually represent solutions to equations. It can make understanding relationships between variables easier and shows how changes in one variable affect another. Visualizing equations can also help solve systems of equations and analyze data trends.
Slope and Y-Intercept
The slope and y-intercept are key components of a linear equation, especially in the slope-intercept form \( y = mx + b \). Knowing these two elements can tell you much about a line:
  • Slope \( (m) \): This value indicates how much \( y \) changes for a unit change in \( x \). The slope \( \frac{6}{5} \), for instance, means that as \( x \) increases by 1, \( y \) increases by \( 1.2 \). Slopes can be positive, negative, zero, or undefined:
    • Positive: \( m > 0 \), line goes upwards.
    • Negative: \( m < 0 \), line goes downwards.
    • Zero: \( m = 0 \), line is horizontal.
    • Undefined: vertical lines, \( x = k \). No y-intercept.
  • Y-Intercept \( (b) \): This is the point where the line touches the y-axis. In \( y = \frac{6}{5}x + 4 \), the y-intercept is 4, marking the point \( (0, 4) \).
Together, these values provide a complete picture of the linear relationship, aiding both predictive and visual understanding in mathematical and real-world contexts.

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Most popular questions from this chapter

Make Sense? During the winter, you program your home thermostat so that at midnight, the temperature is \(55^{\circ} .\) This temperature is maintained until 6 a.m Then the house begins to warm up so that by 9 a.m the temperature is \(65^{\circ} .\) At 6 p.m the house begins to cool. By 9 p.m the temperature is again \(55^{\circ}\). The graph illustrates home temperature, \(f(t),\) as a function of hours after midnight, t. (Graph can't copy) Determine whether each statement makes sense or does not make sense, and explain your reasoning. If the statement makes sense, graph the new function on the domain \([0,24]\). If the statement does not make sense, correct the function in the statement and graph the corrected function on the domain \([0,24]\) I decided to keep the house \(5^{\circ}\) cooler than before, so I reprogrammed the thermostat to \(y=f(t)-5\)

$$\text { Solve for } y: 3 x+2 y-4=0$$

a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1, g(x)=f(2 x), h(x)=f(3 x),\) and \(k(x)=f(4 x)\) in the same viewing rectangle. c. Describe the relationship among the graphs of \(f, g, h\) and \(k,\) with emphasis on different values of \(x\) for points on all four graphs that give the same \(y\) -coordinate. d. Generalize by describing the relationship between the graph of \(f\) and the graph of \(g,\) where \(g(x)=f(c x)\) for \(c>1\) e. Try out your generalization by sketching the graphs of \(f(c x)\) for \(c=1, c=2, c=3,\) and \(c=4\) for a function of your choice.

If you are given a function's graph, how do you determine if the function is even, odd, or neither?

Suppose that a function \(f\) whose graph contains no breaks or gaps on \((a, c)\) is increasing on \((a, b),\) decreasing on \((b, c)\) and defined at \(b\). Describe what occurs at \(x-b\). What does the function value \(f(b)\) represent?
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