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Chapter 3: Problem 19

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.

Short Answer

Expert verified
The solution to the equation is \(x = \frac{7}{2}\), and substituting it back confirms the result.

Step by step solution

01

Distribute the 2

Start by distributing the 2 to both terms inside the parentheses in the equation \(2(x-6)=-5\). This results in: \(2x - 12 = -5\).
02

Add 12 to Both Sides

Add 12 to both sides of the equation to isolate the term with \(x\). So, \(2x - 12 + 12 = -5 + 12\) simplifies to \(2x = 7\).
03

Divide Both Sides by 2

Now, divide both sides by 2 to solve for \(x\): \(\frac{2x}{2} = \frac{7}{2}\). Thus, \(x = \frac{7}{2}\).
04

Verification by Substitution

To verify the solution, substitute \(x = \frac{7}{2}\) back into the original equation. The original equation is \(2(x-6) = -5\). Substitute \(x\) with \(\frac{7}{2}\): \(2\left(\frac{7}{2} - 6\)\right). This simplifies to \(2\left(\frac{7}{2} - \frac{12}{2}\right) = 2\left(-\frac{5}{2}\right) = -5\), which matches the original right side of the equation.

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Key Concepts

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

Equation Solving
Equation solving is all about finding the value of the variable that makes the equation true. In our exercise, we are tasked with solving the equation \(2(x-6)=-5\). The goal is to isolate \(x\) on one side.To solve equations systematically:
  • Follow operations that reverse the actions given in the equation.
  • Start by addressing multiplication or division and progress to addition or subtraction.
By distributing and then isolating the variable \(x\) through inverse operations, you will find that \(x = \frac{7}{2}\) is the solution. Understanding each step allows us to systematically handle even more complex equations.
Distributive Property
The distributive property is an essential concept in algebra. It's a rule that helps us multiply a single term with each term inside a set of parentheses. In the equation \(2(x-6) = -5\), we apply the distributive property by multiplying \(2\) by both \(x\) and \(-6\), so:
  • \(2 \times x = 2x\)
  • \(2 \times (-6) = -12\)
This simplifies our equation to \(2x - 12 = -5\).
By using the distributive property, we transform the equation into a simpler form that makes it easier to isolate and solve for the variable. This principle is vital not only for solving simple equations but also in simplifying complex expressions.
Verification by Substitution
Verification by substitution confirms the solution is correct. Once we find \(x = \frac{7}{2}\), we plug it back into the original equation to ensure it satisfies the equation's conditions.Here's how it works:- Substitute \(x\) back into the equation \(2(x-6)\).- Replace \(x\) with \(\frac{7}{2}\): \(2\left(\frac{7}{2} - 6\right)\).- Simplify the expression: \(2 \times \left(-\frac{5}{2}\right) = -5\).We find that our calculated left side equals the right side of the original equation \(-5\).
This step reaffirms that our solution is correct. Verification is crucial as it provides confidence in your solution's accuracy and helps catch potential errors.

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Most popular questions from this chapter

Suppose a new small-business computer system costs \(\$ 5,400\). Every year its value drops by \(\$ 525\). a. Define variables and write an equation modeling the value of the computer in any given year. b. What is the rate of change, and what does it mean in the context of the problem? c. What is the \(y\)-intercept, and what does it mean in the context of the problem? d. What is the \(x\)-intercept, and what does it mean in the context of the problem?

Draw and label a coordinate plane with each axis scaled from \(-10\) to 10 . a. Represent each point named with a dot, and label it using its letter name. \(\begin{array}{llll}A(3,-2) & B(-8,1.5) & C(9,0) & D(-9.5,-3) \\ F(1,-1) & G(0,-6.5) & H(2.5,3) & I(-6,7.5)\end{array}\) b. List the points in Quadrant I, Quadrant II, Quadrant III, and Quadrant IV. Which points are on the \(x\)-axis? Which points are on the \(y\)-axis? c. Explain how to tell which quadrant a point will be in by looking at the coordinates. Explain how to tell if a point lies on one of the axes.

Solve each equation using the method of your choice. Then use a different method to verify your solution. a. \(14 x=63\) b. \(-4.5 x=18.6\) c. \(8=6+3 x\) d. \(5(x-7)=29\) e. \(3(x-5)+8=12\)

Decide whether each expression is positive or negative without using your calculator. Then check your answer with your calculator. a. \(-35(44)+23\) b. \((-14)(-36)-32\) c. \(25-\frac{152}{12}\) d. \(50-23(-12)\) e. \(\frac{-12-38}{15}\) f. \(24(15-76)\)

For each table, write a formula for list L2 in terms of list L1. $$ \begin{aligned} &\text { a. }\\\ &\begin{array}{|c|c|} \hline \text { L1 } & \text { L2 } \\ \hline 0 & -5.7 \\ \hline 1 & -3.4 \\ \hline 2 & -1.1 \\ \hline 3 & 1.2 \\ \hline 4 & 3.5 \\ \hline 5 & 5.8 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { b. }\\\ &\begin{array}{|r|r|} \hline \text { L1 } & \text { L2 } \\ \hline-3 & 19 \\ \hline-1 & 3 \\ \hline 0 & -5 \\ \hline 2 & -21 \\ \hline 5 & -45 \\ \hline 6 & -53 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { c. }\\\ &\begin{array}{|r|c|} \hline \text { L1 } & \text { L2 } \\ \hline 3 & 13.5 \\ \hline-2 & 11 \\ \hline-9 & 7.5 \\ \hline 0 & 12 \\ \hline 6 & 15 \\ \hline-5 & 9.5 \\ \hline \end{array} \end{aligned} $$
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