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Chapter 4: Problem 17

In Exercises 17-22, complete the table of values. Then plot the solution points on a rectangular coordinate system. $$ \begin{array}{|l|l|l|l|l|l|} \hline x & -2 & 0 & 2 & 4 & 6 \\ \hline y=3 x-4 & & & & & \\ \hline \end{array} $$

Short Answer

Expert verified
The y-values to complete the table are -10, -4, 2, 8 and 14. After placing these points on the rectangular coordinate system, a straight line passing through those points will be visible, which represents the equation \(y = 3x-4\).

Step by step solution

01

Calculate the y values

For each x number in the table, substitute it into the equation \(y = 3x - 4\) to calculate the corresponding y values. For example, when \(x = -2\), \(y = 3(-2) - 4 = -10\).
02

Complete the table

Continue with the substitution process for the remaining \(x\) values. The y-values for \(x = 0\), \(x = 2\), \(x = 4\), and \(x = 6\) will be \(-4\), \(2\), \(8\), \(14\) respectively.
03

Plot the points on the rectangular coordinate system

Each pair \((x, y)\) will serve as coordinates in a 2-dimensional coordinate system. Plot these points on the graph. The points are \((-2, -10)\), \((0, -4)\), \((2, 2)\), \((4, 8)\), \((6, 14)\). Draw the line that passes through the points.

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Key Concepts

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

Rectangular Coordinate System
Understanding the rectangular coordinate system is fundamental for graphing equations. It consists of two perpendicular number lines, the vertical 'y-axis' and the horizontal 'x-axis', which intersect at a point called the origin (0, 0). Every point in the plane is determined by an ordered pair of numbers written as \( (x, y) \). The 'x' value indicates the position along the x-axis, while the 'y' value indicates the position along the y-axis.

In your exercise, you would begin by finding the coordinates through substituting 'x' values into the given equation and then plot these points on the x- and y-axes of your graph.
Substitution Method
The substitution method is a way of finding the exact y-values for a set of x-values in an equation. The process involves replacing each 'x' in the equation with an actual numerical value, then solving for 'y'. The resulting pairs of \(x, y\) values form the solution that fits the given linear equation.

In the given exercise, you substitute values such as -2, 0, 2, 4, and 6 into the equation \(y = 3x - 4\) to find the corresponding y-values. This method provides the precise points that you will later plot on your graph.
Linear Functions
Linear functions are algebraic equations that make straight lines when graphed on a coordinate plane. The general form of a linear equation is \(y = mx + b\), where 'm' represents the slope鈥攖he angle and direction of the line鈥攁nd 'b' signifies the y-intercept, the point where the line crosses the y-axis.

Key Characteristics of Linear Functions:

  • The graph of a linear function is always a straight line.
  • Each input 'x' has exactly one output 'y'.
  • The slope 'm' tells us how steep the line is.
Your exercise features the linear function \(y = 3x - 4\), where 3 is the slope and -4 the y-intercept.
Graphing Linear Equations
Graphing linear equations involves plotting the solutions on a graph with a rectangular coordinate system. After obtaining the set of \(x, y\) pairs using the substitution method, you plot each point on the graph. These points should align and create a straight line when connected.

The graph visually represents all the solutions to the linear equation, allowing you to see the relationship between x and y. Following through with the exercise, after plotting points such as \( (2, 2) \) and \( (4, 8) \), and drawing the line through them, you complete the visual representation of the linear function.

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Most popular questions from this chapter

The table shows the numbers \(y\) (in millions) of adults (over 18 years of age) never married in the United States for the years 2006 through \(2011 .\) $$ \begin{array}{|c|c|} \hline \text { Year } & \boldsymbol{y} \\ \hline 2006 & 55.3 \\ \hline 2007 & 56.1 \\ \hline 2008 & 58.3 \\ \hline 2009 & 59.1 \\ \hline 2010 & 61.5 \\ \hline 2011 & 63.3 \\ \hline \end{array} $$ A model for this data is \(y=1.63 t+45.1\), where \(t\) is the year, with \(t=6\) corresponding to 2006 . (Source: U.S. Census Bureau) (a) Plot the data and graph the model on the same set of coordinate axes. (b) Use the model to predict the number of adults over the age of 18 in 2020 who will never have married.

You purchase a boat for $$\$ 25,000$$. After 1 year, its depreciated value is $$\$ 22,700$$. The depreciation is linear. (a) Write a linear model that relates the value \(V\) of the boat to the time \(t\) in years. (b) Use the model to estimate the value of the boat after 3 years.

In Exercises \(61-64\), solve for \(y\) in terms of \(x\). $$ 4 x-5 y=-2 $$

In Exercises 1-8, plot the points on a rectangular coordinate system. $$ (3,2),(-4,2),(2,-4) $$

You run and walk on a trail that is 6 miles long. You run 4 miles per hour and walk 3 miles per hour. Let \(y\) be the number of hours you walk and let \(x\) be the number of hours you run. (a) Write an equation that relates the number of hours you run and the number of hours you walk to the total length of the trail. (b) Sketch the graph of the equation. (c) What is the \(y\)-intercept of the graph, and what does it represent in the context of the problem?
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