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Chapter 3: Problem 12

An equation can have the variable on both sides. In these cases you can maintain the balance by eliminating the \(x\) 's from one of the sides before you begin undoing. a. Copy and complete this table to solve the equation. (a) b. Show the steps used to solve \(5 x-4=2 x+5\) using the balancing method. Substitute your solution into the original equation to check your answer.

Short Answer

Expert verified
The solution is \(x = 3\).

Step by step solution

01

Understanding the Equation

We are given the equation \(5x - 4 = 2x + 5\). Our goal is to solve for \(x\) by maintaining the balance between both sides of the equation.
02

Eliminate Variable from One Side

Subtract \(2x\) from both sides of the equation to eliminate the \(x\) term from the right side. \[(5x - 2x) - 4 = 2x - 2x + 5\]Simplifying, we get:\[3x - 4 = 5\]
03

Isolate the Variable Term

Add 4 to both sides of the equation to isolate the term containing \(x\) on the left side. \[3x - 4 + 4 = 5 + 4\]This simplifies to:\[3x = 9\]
04

Solve for x

Divide both sides of the equation by 3 to solve for \(x\).\[\frac{3x}{3} = \frac{9}{3}\]So, \(x = 3\).
05

Substitute Solution Back into Original Equation

To verify, substitute \(x = 3\) back into the original equation \(5x - 4 = 2x + 5\):LHS: \(5(3) - 4 = 15 - 4 = 11\)RHS: \(2(3) + 5 = 6 + 5 = 11\)Both sides are equal, confirming our solution is correct.

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Key Concepts

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

Understanding Balancing Method
When we talk about the balancing method in equation solving, we're trying to maintain equilibrium, like a balanced scale. Think of an equation as a scale, where both sides should weigh the same. This concept is crucial because whatever operation you perform on one side of the equation, you must perform the same operation on the other side to keep it balanced.
  • For example, if you add 3 to one side, you must add 3 to the other side.
  • This ensures that the fundamental equality of the equation is preserved.
In the given equation, the balancing method first involves ensuring that we perform identical operations on both sides to simplify the expression. This is especially important when variables appear on both sides.
Technique of Variable Isolation
Variable isolation is about getting the variable, usually denoted as \(x\), alone on one side of the equation so that you can easily find its value.To isolate the variable, you need to look for the variable term and consider the operations being performed on it. These operations could be addition, subtraction, multiplication, or division. Whatever is affecting the variable must be undone, so the variable stands alone.
  • For instance, if your equation is \(3x - 4 = 5\), you start by undoing any addition or subtraction involving \(x\).
  • You would add 4 to both sides to get \(3x = 9\).
  • Next, divide by the coefficient of \(x\) (which is 3 in this case) to isolate \(x\).
This process is like peeling away layers from the equation to reveal the core value of \(x\).
Step-by-step Solution Approach
Breaking down the problem into a step-by-step solution helps clarify each move that gets us closer to finding \(x\).
  • Start by simplifying each side of the equation as much as possible.
  • Eliminate terms involving the variable from one side so that you have the variable only on one side of the equation.
  • Isolate the variable by reversing the operations affecting it, allowing you to solve for the variable.
This methodical breakdown reduces complexities and allows a clear path to the solution. Following each step carefully ensures no mistakes are made, and every action is logical and effective.
Checking with Substitution
Substitution check is like a test drive for your solution. It's about ensuring that the value of \(x\) you've obtained is correct by plugging it back into the original equation.Take your calculated value of \(x\) and substitute it wherever \(x\) appears in the original equation.
  • Calculate the left-hand side (LHS) of the equation with this value.
  • Then, calculate the right-hand side (RHS).
  • If both sides are equal after substitution, your solution is verified as correct.
For our problem, substituting \(x = 3\) should make both sides equal 11. This step confirms your solution is accurate, rounding off the problem-solving process with certainty.

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Most popular questions from this chapter

APPLICATION Amber makes \(S 6\) an hour at a sandwich shop. She wants to know how many hours she needs to work to save \(\$ 500\) in her bank account. On her first paycheck, she notices that her net pay is about \(75 \%\) of her gross pay. a. How many hours must she work to earn \(\$ 500\) in gross pay? b. How many hours must she work to earn \(\$ 500\) in net pay?

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. Bagel Store \begin{tabular}{|l|l|l|l|l|l|l|} \hline Bagels & 13 & 26 & 39 & 52 & 65 & 78 \\ \hline Cost & & & & & & \\ \hline \end{tabular} Grocery Store \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|} \hline Bagels & 6 & 12 & 18 & 24 & 30 & 36 & 42 & 48 & 54 & 60 \\ \hline Cost & & & & & & & & & & \\ \hline \end{tabular} 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: \(6.49\) ????A Ans 2 ? ?ten 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?

Consider the equation \(2(x-6)=-5\). a. Solve the equation. b. Show how you can check your result by substituting it into the original equation.

Write the equations for linear relationships that have these characteristics. a. The output value is equal to the input value. b. The output value is 3 less than the input value. c. The rate of change is \(2.3\) and the \(y\)-intercept is \(-4.3\). d. The graph contains the points \((1,1),(2,1)\), and \((3,1)\).

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? b. \(15 \%\) of \(18.95\) is what number? c. What percent of 64 is 326 ? d. \(10 \%\) of what number is 40 ?
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