/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 (a) Solve the equation \(3 x+2 y... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 8

(a) Solve the equation \(3 x+2 y=6\) for \(y\) and then complete the following table. $$\begin{array}{lccccc} \hline x & -4 & -2 & 0 & 2 & 4 \\ \hline y & & & & & \\ \hline \end{array}$$ (b) Use your table from part (a) to graph the equation \(3 x+2 y=6\)

Short Answer

Expert verified
Equation for y: \( y = 3 - \frac{3}{2}x \). Completed table: y values are 9, 6, 3, 0, -3. Graph by plotting these points.

Step by step solution

01

Rewriting the Equation in Terms of y

To solve for \( y \) in the equation \( 3x + 2y = 6 \), we first need to isolate \( y \) on one side of the equation. Start by subtracting \( 3x \) from both sides to get:\[ 2y = 6 - 3x \] Next, divide every term by 2 to solve for \( y \):\[ y = 3 - \frac{3}{2}x \] This is the equation in terms of \( y \).
02

Calculating y Values for Given x Values

Now, we will substitute the given values of \( x \) into the equation \( y = 3 - \frac{3}{2}x \) to find the corresponding \( y \) values. - For \( x = -4 \): \( y = 3 - \frac{3}{2}(-4) = 3 + 6 = 9 \) - For \( x = -2 \): \( y = 3 - \frac{3}{2}(-2) = 3 + 3 = 6 \) - For \( x = 0 \): \( y = 3 - \frac{3}{2}(0) = 3 \) - For \( x = 2 \): \( y = 3 - \frac{3}{2}(2) = 3 - 3 = 0 \) - For \( x = 4 \): \( y = 3 - \frac{3}{2}(4) = 3 - 6 = -3 \)
03

Completing the Table

Using the \( y \) values calculated in the previous step, we can complete the table:\[ \begin{array}{lccccc}\hline x & -4 & -2 & 0 & 2 & 4 \\hline y & 9 & 6 & 3 & 0 & -3 \\hline\end{array} \]
04

Plotting Points for the Graph

Now that we have both \( x \) and \( y \) values, we can plot them on a graph. The points are:- \((-4, 9)\),- \((-2, 6)\),- \((0, 3)\),- \((2, 0)\),- \((4, -3)\). These will be plotted on a coordinate plane, drawing a straight line through all the points.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Graphing
Graphing is the process of plotting points, shapes, and lines on a graph to visually represent an equation or data set. When you graph a linear equation, you are essentially drawing the line that shows all solutions to the equation. Each point on the line is a solution to the equation.

To graph the equation from the exercise, we first solved for different values of \( y \) based on given values of \( x \). These values form points on the graph. In this exercise, after calculating the corresponding \( y \) values for \( x = -4, -2, 0, 2, \) and \( 4 \), we obtained the points:
  • \((-4, 9)\)
  • \((-2, 6)\)
  • \((0, 3)\)
  • \((2, 0)\)
  • \((4, -3)\)
Now, by placing these points on the graph, you can draw a straight line that passes through all of them. This line is the graphical representation of the equation \(3x + 2y = 6\). It gives a clear and intuitive visual of how the equation behaves and what solutions it represents.
Coordinate Plane
The coordinate plane is a two-dimensional surface on which you can plot points, lines, and curves. It consists of two axes that intersect at a point called the origin. The horizontal axis is known as the x-axis, and the vertical axis is the y-axis.

The origin, where the axes meet, is the point \((0, 0)\). Every point on the coordinate plane is identified by an ordered pair \((x, y)\), where \(x\) represents the distance along the x-axis and \(y\) represents the distance along the y-axis. This system allows for precise positions of points, lines, and shapes on the plane.

In this exercise, we used the coordinate plane to graph the equation \(3x + 2y = 6\). Plotting the calculated points \((-4, 9), (-2, 6), (0, 3), (2, 0), (4, -3)\) on the coordinate plane helps us understand the solutions to the equation. It visually illustrates the relationship between the \(x\) and \(y\) values as a straight line, revealing the linearity and slope of the equation.
Solving for y
Solving an equation for \(y\) means manipulating the equation to express the variable \(y\) in terms of \(x\) and other constants. This transformation makes it easier to understand and use the equation, especially for graphing.

In the context of linear equations like the one in the exercise, transforming the equation into the form \(y = mx + b\) reveals its slope-intercept form, which is handy for graphing. Here, \(m\) is the slope showing how steep the line is, while \(b\) is the y-intercept, the point where the line crosses the y-axis.

For the equation \(3x + 2y = 6\), we isolated \(y\) to get \(y = 3 - \frac{3}{2}x\). This equation shows that for every unit increase in \(x\), \(y\) decreases by \(\frac{3}{2}\), indicating a negative slope. By solving for \(y\), we've made it easier to plug in values of \(x\) and find corresponding \(y\) values for plotting on the coordinate plane.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x-5| \geq 2$$

Determine the center and the radius for the circle. Also, find the \(y\) -coordinates of the points (if any) where the circle intersects the \(y\) -axis. $$x^{2}+y^{2}+8 x-6 y=-24$$

(a) Use a graphing utility to graph the equation. (b) Use a graphing utility, as in Example \(5,\) to estimate to one decimal place the \(x\) -intercepts. (c) Use algebra to determine the exact values for the \(x\) -intercepts. Then use a calculator to check that the answers are consistent with the estimates obtained in part (b). $$y=3 x^{3}+5 x^{2}+x$$

(a) Sketch the line \(y=\frac{1}{2} x-5\) and the point \(P(1,3) .\) Follow parts (b)-(d) to calculate the perpendicular distance from point \(P(1,3)\) to the line. (b) Find an equation of the line that passes through \(P(1,3)\) and is perpendicular to the line \(y=\frac{1}{2} x-5\) (c) Find the coordinates of the point where these two lines intersect. Hint: From intermediate algebra, to find where two lines \(y=m x+b\) and \(y=M x+B\) intersect, set the expressions \(m x+b\) and \(M x+B\) equal to each other, and solve for \(x\) (d) Use the distance formula to find the perpendicular distance from \(P(1,3)\) to the line \(y=\frac{1}{2} x-5\)

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$\left|x+\frac{1}{3}\right|<\frac{3}{2}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.