/*! 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 33 Graph the given equation. $$ ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 33

Graph the given equation. $$ 3 x=8 $$

Short Answer

Expert verified
The short answer: Solve the equation \(3x = 8\) to find \(x = \frac{8}{3}\). Graph the solution on a number line by placing a filled circle at the point \(\frac{8}{3}\).

Step by step solution

01

Solve the equation for x

Divide both sides of the equation by 3 to isolate x: \(x = \frac{8}{3}\)
02

Prepare the number line

Draw a horizontal line and mark the position of \(\frac{1}{3}\), \(\frac{2}{3}\), \(\frac{8}{3}\), and \(\frac{9}{3}\) on it. This will help us visualize the solution in context.
03

Plot the solution on the number line

Place a filled circle at the point \(\frac{8}{3}\) on the number line. This represents the solution \(x = \frac{8}{3}\).
04

Check the solution

To ensure our graph is correct, check that when you substitute \(x = \frac{8}{3}\) back into the equation, the equation is true: \(3 (\frac{8}{3}) = 8\) The graph of the equation \(3x = 8\) is a number line with a filled circle at the point \(x = \frac{8}{3}\).

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Key Concepts

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

Number Line Representation
Understanding how to use a number line is crucial when graphing solutions to linear equations. A number line is a simple visual tool that helps us easily spot and verify solutions.

To create a number line, draw a straight, horizontal line and label it with tick marks to indicate units or fractions relevant to the problem. In our exercise, we specifically marked the fractions \(\frac{1}{3}\), \(\frac{2}{3}\), and importantly, \(\frac{8}{3}\). These fractions help us precisely visualize where the solution \(x = \frac{8}{3}\) lies.
  • Place tick marks at equal intervals to represent simple fractions.
  • Identify and highlight the solution, here at \(\frac{8}{3}\), using a filled circle or dot to represent the exact value.
This visual representation not only makes the solution more tangible but also reinforces the accuracy of solving linear equations.
Solving Linear Equations
Linear equations often come in the form \(ax = b\), where \(a\) and \(b\) are constants. Our goal is to find \(x\). For instance, to solve the equation \(3x = 8\), we must isolate \(x\) on one side of the equation.

Here鈥檚 a simple solution:
  • Divide both sides of the equation by the coefficient of \(x\), which means dividing \(8\) by \(3\) to solve for \(x\).
  • This gives us \(x = \frac{8}{3}\).
This process demonstrates the core principle of maintaining equality. Dividing both sides of an equation by the same number ensures the balance remains intact, revealing the solution as \(x = \frac{8}{3}\). Recognizing and applying these steps repeatedly enhances our ability to solve any linear equation efficiently.
Equation Verification
Verifying your solution is a critical step in solving linear equations. By substituting the value back into the original equation, you confirm the accuracy of your solution.

For our problem, substitute \(x = \frac{8}{3}\) back into the original equation:
  • Replace \(x\) with \(\frac{8}{3}\), giving you: \(3(\frac{8}{3})\).
  • When simplified, this calculation equals \(8\), matching the equation's right side.
This check is quick and invaluable. It reassures us that the calculated solution \(x = \frac{8}{3}\) satisfies the initial equation \(3x = 8\), leaving no room for doubt. Always ensure your solution is genuine by reconciling it with the original problem.

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Most popular questions from this chapter

You have ascertained that a table of values of \(x\) and \(y\) corresponds to a linear function. How do you find an equation for that linear function?

If the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) lie on a straight line, what can you say about the regression line associated with these points?

Use technology to compute the sum-ofsquares error (SSE) for the given set of data and linear models. Indicate which linear model gives the better fit. $$ (0,1),(1,1),(2,2) ; \quad \text { a. } y=0.4 x+1.1 \quad \text { b. } y=0.5 x+0.9 $$

Find a linear equation whose graph is the straight line with the given properties. Through \((3.5,-10)\) and increasing at a rate of 1 unit of \(y\) per 2 units of \(x\)

Find a linear equation whose graph is the straight line with the given properties. Through \((1 / 3,-1)\) and parallel to the line \(3 x-4 y=8\)
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