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Chapter 3: Problem 5

Write the equations for linear relationships that have these characteristics. a. The output value is equal to the input value. b. The output value is 3 less than the input value. c. The rate of change is \(2.3\) and the \(y\)-intercept is \(-4.3\). d. The graph contains the points \((1,1),(2,1)\), and \((3,1)\).

Short Answer

Expert verified
a) \( y = x \); b) \( y = x - 3 \); c) \( y = 2.3x - 4.3 \); d) \( y = 1 \).

Step by step solution

01

Definition of a linear relationship

A linear relationship between two variables can be represented by an equation of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Our task is to find the appropriate values of \( m \) and \( b \) for each characteristic provided in the question.
02

Solve for characteristic (a)

For characteristic (a), where the output value is equal to the input value, the equation is simply \( y = x \). This means the slope \( m = 1 \) and the y-intercept \( b = 0 \).
03

Solve for characteristic (b)

For characteristic (b), where the output value is 3 less than the input value, the equation is \( y = x - 3 \). Here, the slope \( m = 1 \) and the y-intercept \( b = -3 \).
04

Solve for characteristic (c)

In characteristic (c), we are given a rate of change, which is the slope \( m = 2.3 \), and a y-intercept \( b = -4.3 \). Therefore, the equation is \( y = 2.3x - 4.3 \).
05

Solve for characteristic (d)

In characteristic (d), the graph contains the points \((1,1), (2,1),\) and \((3,1)\). All points have the same y-value, indicating a horizontal line with a slope \( m = 0 \). Therefore, the equation is \( y = 1 \), which implies the line is parallel with the x-axis at \( y = 1 \).

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Key Concepts

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

Linear Equations
Linear equations are at the heart of understanding linear relationships. A linear equation is any equation that, when graphed, forms a straight line. This type of equation is often written in the form of \[ y = mx + b \] where:
  • \( y \) represents the dependent variable (output).
  • \( x \) is the independent variable (input).
  • \( m \) stands for the slope of the line.
  • \( b \) is the y-intercept.
The relationship defined by a linear equation is crucial because it helps us predict the value of one variable based on another. By understanding how the input variable \( x \) relates linearly to the output variable \( y \), you can interpret and predict the behavior of the system you're examining. These relationships can model real-world situations, like calculating costs or predicting trends.
Slope
The slope in a linear equation, denoted as \( m \), measures the steepness and direction of the line. It tells you how much \( y \) changes for a unit change in \( x \). If you see a high slope, it means the line is steep, and if it鈥檚 low, the line is relatively flat. Here鈥檚 how you might think about slope:
  • If \( m > 0 \), the line rises from left to right, indicating a positive relationship between \( x \) and \( y \).
  • If \( m < 0 \), the line falls from left to right, indicating a negative relationship.
  • If \( m = 0 \), the line is horizontal, showing no change in \( y \) regardless of \( x \).
To calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Grasping the concept of slope will help you understand and analyze the behavior of any linear equation.
Y-intercept
The y-intercept, denoted as \( b \) in the equation \( y = mx + b \), is the point where the line crosses the y-axis. This occurs when the value of \( x \) is zero. The y-intercept provides a starting point for graphing a line on a coordinate plane.Understanding the y-intercept is vital because it represents the initial value of \( y \) when \( x \) is zero:
  • A positive y-intercept means the line crosses above the origin.
  • A negative y-intercept means it crosses below the origin.
  • If \( b = 0 \), the line passes through the origin. This would mean that at the initial state (where \( x = 0 \)), the output \( y = 0 \) as well.
In practical terms, the y-intercept could be thought of as the starting balance of an account before deposits (changes) begin, or the original height of a tree before it starts growing. Recognizing and calculating the y-intercept helps you understand where your linear graph starts on the y-axis.

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Most popular questions from this chapter

Solve each equation symbolically using the balancing method. a. \(3+2 x=17\) b. \(0.5 x+2.2=101.0\) c. \(x+307.2=2.1\) d. \(2(2 x+2)=7\) e. \(\frac{4+0.01 x}{6.2}-6.2=0\) (d)

Mini-Investigation A solution to the equation \(-10+3 x=5\) is shown below. $$ \begin{aligned} -10+3 x &=5 \\ 3 x &=15 \\ x &=5 \end{aligned} $$ a. Describe the steps that transform the original equation into the second equation and the second equation into the third (the solution). b. Graph \(\mathrm{Y}_{1}=-10+3 x\) and \(\mathrm{Y}_{2}=5\), and trace to the lines' intersection. Write the coordinates of this point. c. Graph \(\mathrm{Y}_{1}=3 x\) and \(\mathrm{Y}_{2}=15\), and trace to the lines' intersection. Write the coordinates of this point. d. Graph \(\mathrm{Y}_{1}=x\) and \(\mathrm{Y}_{2}=5\), and trace to the lines' intersection. Write the coordinates of this point. e. What do you notice about your answers to \(8 \mathrm{~b}-\mathrm{d}\) ? Explain what this illustrates.

On his Man in Motion World Tour in 1987, Canadian Rick Hansen wheeled himself \(24,901.55\) miles to support spinal cord injury research and rehabilitation, and wheelchair sport. He covered 4 continents and 34 countries in two years, two months, and two days. Learn more about Rick's journey with the link. at mww.kevmath.comiDA. a. Find Rick's average rate of travel in miles per day. (Assume there are 365 days in a year and \(30.4\) days in a month.) (Tit) b. How much farther would Rick have traveled if he had continued his journey for another \(1 \frac{1}{2}\) years? c. If Rick continued at this same rate, how many days would it take him to travel 60,000 miles? How many years is that?

You can solve familiar formulas for a specific variable. For example, solving \(A=l w\) for \(l\) you get \(\begin{array}{ll}A=t w & \text { Original equation. } \\ \frac{A}{w}=\frac{i w}{w} & \text { Divide both sides by }{ }^{A} . \\ \frac{A}{w}=l & \text { Reduce, }\end{array}\) You can also write \(t=\frac{\Delta}{w^{-}}\)Now try solving these formulas for the given variable. a. \(\mathrm{C}=2 \pi r\) for \(r\) (a) b. \(A-\frac{1}{2}(h b)\) for \(h\) c. \(P=2(l+w)\) for \(l\) (a) d. \(P=4 s\) for \(s\) e. \(d=r t\) for \(t\) f. \(A=\frac{1}{2} h(a+b)\) for \(h\)

Solve these equations. Tell what action you take at each stage. a. \(144 x=12\) b. \(\frac{1}{6} x+2=8\)
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