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Chapter 5: Problem 151

Solve the equation. Round your answer to three decimal places, if necessary. $$3 x-7=14$$

Short Answer

Expert verified
The solution to the equation is \(x = 7\).

Step by step solution

01

Isolate the Variable

Start by adding 7 to both sides of the equation \(3x - 7 = 14\). It will give \(3x = 21\).
02

Solve for x

Divide both sides of the equation by 3 to solve for \(x\). It will result in \(x = 7\).
03

Verify the Solution

Substitute \(x = 7\) back into the original equation \(3x - 7 = 14\) to verify the solution. It gives \(21 - 7 = 14\) which is correct.

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Key Concepts

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

Solving Linear Equations
Solving linear equations is like solving a puzzle. The goal is to find the value of the variable that makes the equation true. A linear equation is an equation in which each term is either a constant or the product of a constant and a single variable. Equations like \(3x - 7 = 14\) are typical linear equations. Here, our job is to determine the value of \(x\) in a straightforward and methodical way.
\[\]Let's break it down into clear steps:
  • Look at both sides of the equation. Identify the operations applied to the variable you need to solve for.
  • Use inverse operations to find the value of the variable. This means doing the opposite operation to cancel out terms and simplify the equation.
By following these steps, you carefully balance the equation and gradually move the variable to one side while all other numbers go to the other side.
Isolation of Variables
The core goal in simplifying any equation is to isolate the variable. This means getting the variable by itself on one side of the equation. Only then can you clearly see its value or relationship to other numbers. In our sample equation \(3x - 7 = 14\), "isolating" \(x\) involves a couple of key moves:
\[\]
  • Addition or Subtraction: Start by adding or subtracting terms to both sides of the equation that don鈥檛 involve the variable. This helps move constants to the opposite side.
  • Multiplication or Division: Once all additive or subtractive constants are moved, you can multiply or divide to finalize isolation. In our example, after getting \(3x = 21\), divide by 3 to give \(x = 7\).
This method is like peeling an onion, layer by layer until you reach the core variable.
Verification of Solutions
Verifying the solution is the final essential step in solving equations. Even if you think you've found the correct answer, it's crucial to confirm it is correct by substituting the solution back into the original equation.
\[\]Here's how verification works:
  • Take the value you found for the variable and plug it back into the original equation.
  • Calculate each side separately to ensure both sides of the equation are equal.
For instance, with \(x = 7\) in the original equation \(3x - 7 = 14\), substitute 7 for \(x\). You get \(3(7) - 7 = 14\). This equals \(21 - 7 = 14\), confirming that the solution is valid. Verification not only strengthens accuracy but also boosts your confidence in your algebraic skills.

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Most popular questions from this chapter

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=-270^{\circ}$$

A ball that is bobbing up and down on the end of a spring has a maximum displacement of 3 inches. Its motion (in ideal conditions) is modeled by \(y=\frac{1}{4} \cos 16 t, \quad t>0.\) where \(y\) is measured in feet and \(t\) is the time in seconds. (a) Use a graphing utility to graph the function. (b) What is the period of the oscillations? (c) Determine the first time the ball passes the point of equilibrium \((y=0).\)

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=7 \pi / 2$$

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$-0.48$$

True or False Determine whether the statement is true or false. Justify your answer. Simple harmonic motion does not involve a damping factor.
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