/*! 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 10 Solve each equation symbolically... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 10

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

Short Answer

Expert verified
(a) 7; (b) 197.6; (c) -305.1; (d) 0.75; (e) 220

Step by step solution

01

Isolate the Term with the Variable

For equation (a): Start by subtracting 3 from both sides to isolate terms with the variable.\[ 3 + 2x - 3 = 17 - 3 \]This simplifies to:\[ 2x = 14 \]
02

Solve for the Variable by Balancing

For equation (a): Divide both sides of the equation by 2 to solve for \( x \).\[ \frac{2x}{2} = \frac{14}{2} \]Simplifying gives:\[ x = 7 \]
03

Isolate the Term with the Variable

For equation (b): Subtract 2.2 from both sides to isolate the term with the variable.\[ 0.5x + 2.2 - 2.2 = 101.0 - 2.2 \]This simplifies to:\[ 0.5x = 98.8 \]
04

Solve for the Variable by Balancing

For equation (b): Divide both sides of the equation by 0.5 to solve for \( x \).\[ \frac{0.5x}{0.5} = \frac{98.8}{0.5} \]This simplifies to:\[ x = 197.6 \]
05

Isolate the Term with the Variable

For equation (c): Subtract 307.2 from both sides to isolate the variable.\[ x + 307.2 - 307.2 = 2.1 - 307.2 \]This simplifies to:\[ x = -305.1 \]
06

Distribute and Combine Like Terms

For equation (d): Start by distributing the 2 through the parentheses.\[ 2(2x + 2) = 7 \]Expands to:\[ 4x + 4 = 7 \]
07

Isolate the Term with the Variable

For equation (d): Subtract 4 from both sides to get:\[ 4x + 4 - 4 = 7 - 4 \]This simplifies to:\[ 4x = 3 \]
08

Solve for the Variable by Balancing

For equation (d): Divide both sides by 4 to solve for \( x \).\[ \frac{4x}{4} = \frac{3}{4} \]Simplifying gives:\[ x = 0.75 \]
09

Isolate the Term with the Variable

For equation (e): Combine like terms on the left side.\[ 4 + 0.01x - 6.2 = 0 \]Becomes:\[ 0.01x - 2.2 = 0 \]
10

Solve for the Variable by Balancing

For equation (e): Add 2.2 to both sides to isolate the term with the variable.\[ 0.01x - 2.2 + 2.2 = 0 + 2.2 \]This simplifies to:\[ 0.01x = 2.2 \]
11

Solve for the Variable

For equation (e): Divide both sides of the equation by 0.01.\[ \frac{0.01x}{0.01} = \frac{2.2}{0.01} \]Simplifying gives:\[ x = 220 \]

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.

Solving Equations
Solving equations involves finding the value of a variable that makes the equation true. To tackle equations symbolically, you need patience and a systematic approach. The balancing method is a classic technique used in algebra to solve equations. The principle is simple: what you do to one side of the equation, you do to the other, maintaining equilibrium.
  • Start by identifying the operations that are encasing the variable, such as addition or multiplication.
  • Apply inverse operations to strip away these layers and reveal the variable.
  • Keep your goal in mind: isolate the variable to one side of the equation.

For example, with the equation \( 3 + 2x = 17 \), the first step is to remove the 3 by subtracting it from both sides. This step brings us closer to isolating \( x \).
Algebraic Techniques
Algebraic techniques are essential for rearranging equations and simplifying them. One key technique is distributing, which involves applying a multiplier to each term within parentheses. This step simplifies complex expressions, making the equation easier to handle.
Another technique is combining like terms, which means adding or subtracting terms that have the same variable raised to the same power. This process also aids in simplifying the equation. Consider the equation \( 2(2x + 2) = 7 \). Start by distributing 2 to both \( 2x \) and 2, resulting in \( 4x + 4 \). By refining the equation to simpler terms, solving becomes more straightforward.
  • Use the distributive property to expand expressions.
  • Combine like terms to consolidate similar components.
  • Focus on simplifying the equation to a basic form where the variable stands alone.
Isolation of Variables
Isolation of variables is crucial for solving equations effectively. The main objective is to "free" the variable from coefficients and constant terms. This often involves a series of inversions: addition becomes subtraction, multiplication becomes division, and vice versa.
Let's consider the example \( 0.5x + 2.2 = 101.0 \). First, eliminate the constant 2.2 by subtracting it from both sides, resulting in \( 0.5x = 98.8 \). Next, divide by 0.5 to solve for \( x \).
  • Subtract or add constants to "move" them across the equation, targeting the variable.
  • Divide or multiply to get the variable alone.
  • Ensure that the variable appears on one side only for clarity and simplicity.

Isolating variables efficiently is satisfying and often leads to straightforward solutions for equations. Understanding this concept aids significantly in all algebra-related problem solving.

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

At a family picnic, your cousin tells you that he always has a hard time remembering how to compute percents. Write him a note explaining what percent means. Use these problems as examples of how to solve the different types of percent problems, with an answer for each. a. 8 is \(15 \%\) of what number? (a) b. \(15 \%\) of \(18.95\) is what number? c. What percent of 64 is 326 ? d. \(10 \%\) of what number is 40 ?

APPLICATION To plan a trip downtown, you compare the costs of three different parking lots. ABC Parking charges \(\$ 5\) for the first hour and \(\$ 2\) for each additional hour or fraction of an hour. Cozy Car charges \(\$ 3\) per hour or fraction of an hour, and The Corner Lot charges a \(\$ 15\) flat rate for a whole day. a. Make a table similar to the one shown. Write recursive routines to calculate the cost of parking up to 10 hours at each of the three lots. b. Make three different scatter plots on the same pair of axes showing the parking rates at the three different lots. Use a different color for each parking lot. Put the hours on the horizontal axis and the cost on the vertical axis. c. Compare the three scatter plots. Under what conditions is each parking lot the best deal for your trip? Use the graph to explain. d. Would it make sense to draw a line through each set of points? Explain why or why not.

Sketch a graph of a walk starting at the l-meter mark and walking away from the sensor at a constant rate of \(0.5\) meter per second.

A bicyclist, \(1 \mathrm{mi}(5280 \mathrm{ft})\) away, pedals toward you at a rate of \(600 \mathrm{ft} / \mathrm{min}\) for \(3 \mathrm{~min}\). The bicyclist then pedals at a rate of \(1000 \mathrm{ft} / \mathrm{min}\) for the next \(5 \mathrm{~min}\). a. Describe what you think the plot of (time, distance from you) will look like. (a) b. Graph the data using 1 min intervals for your plot. (a) c. Invent a question about the situation, and use your graph to answer the question.

The local bagel store sells a baker's dozen of bagels for \(\$ 6.49\), while the grocery store down the street sells a bag of 6 bagels for \(\$ 2.50\). a. Copy and complete the tables showing the cost of bagels at the two stores. b. Graph the information for each market on the same coordinate axes. Put bagels on the horizontal axis and cost on the vertical axis. c. Find equations to describe the cost of bagels at each store. d. How much does one bagel cost at each store? How do these cost values relate to the equations you wrote in \(15 \mathrm{c}\) ? e. Looking at the graphs, how can you tell which store is the cheaper place to buy bagels? f. Bernie and Buffy decided to use a recursive routine to complete the tables. Bernie used this routine for the bagel store: \(6.49\) ??? Ans - 2 E?m Buffy says that this routine isn't correct, even though it gives the correct answer for 13 and 26 bagels. Explain to Bernie what is wrong with his recursive routine. What routine should he use?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

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.