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Chapter 4: Problem 2

Solve each equation by backtracking. (Backtrack mentally if you can.) Check your solutions. $$ 2(n-5)=7 $$

Short Answer

Expert verified
n = \frac{17}{2}

Step by step solution

01

- Distribute

Distribute the 2 across the expression inside the parenthesis.\[ 2(n - 5) = 7 \] becomes\[ 2n - 10 = 7 \]
02

- Add 10 to both sides

Add 10 to both sides to isolate the term with the variable.\[ 2n - 10 + 10 = 7 + 10 \]This simplifies to\[ 2n = 17 \]
03

- Divide both sides by 2

Divide both sides by 2 to solve for n.\[ \frac{2n}{2} = \frac{17}{2} \]This simplifies to\[ n = \frac{17}{2} \]
04

- Verify the solution

Substitute \( n = \frac{17}{2} \) back into the original equation to check if it's correct.\[ 2\left(\frac{17}{2} - 5\right) = 7 \]Simplify inside the parenthesis:\[ \frac{17}{2} - 5 = \frac{17}{2} - \frac{10}{2} = \frac{7}{2} \]Now multiply by 2:\[ 2 \times \frac{7}{2} = 7 \]Since both sides equal 7, the solution is verified.

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Key Concepts

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

Solving Linear Equations with the Backtracking Method
The backtracking method in algebra is a useful problem-solving strategy for equations. It involves reversing the steps to isolate the variable to solve the equation. Here鈥檚 a step-by-step guide to solve the given linear equation using backtracking.

Consider the equation: \[ 2(n-5) = 7 \]. To solve it:

  • Step 1: Distribute the 2 across the parenthesis.
  • Step 2: Move the numbers to isolate the variable term.
  • Step 3: Divide to get the variable alone.
  • Step 4: Verify the solution to ensure its correctness.
Distribution in Algebra
Distribution is a fundamental property in algebra, which involves multiplying a single term across terms within parentheses. It helps in simplifying and solving equations.

For the equation \[ 2(n-5) = 7 \], we use distribution as follows:

\[ 2 \times n - 2 \times 5 = 7 \]

This simplifies to \[ 2n - 10 = 7 \].

Distribution helps break down complex parts of an equation, making it easier to handle.

Remember, always distribute multiplication over addition or subtraction inside parentheses first.
Isolating the Variable
Isolating the variable means moving all terms containing the variable to one side of the equation and constants to the other.

For \[ 2n - 10 = 7 \]:

We add 10 to both sides to move the constant to the right:

\[ 2n - 10 + 10 = 7 + 10 \]

This simplifies to \[ 2n = 17 \].

Next, divide by 2 to isolate 'n':

\[ \frac{2n}{2} = \frac{17}{2} \]

This results in \[ n = \frac{17}{2} \].

Isolating the variable is crucial for solving linear equations, leading directly to the solution.
Verification of Solutions
Verification ensures the correctness of your solution by substituting it back into the original equation.

Start with the found solution \[ n = \frac{17}{2} \] and check if it satisfies the original equation \[ 2(n-5) = 7 \]:

Substitute \[ n = \frac{17}{2} \] into the equation:

\[ 2\bigg(\frac{17}{2} - 5\bigg) = 7 \]

Simplify inside the parentheses:\br>
\[ 2\bigg(\frac{17}{2} - \frac{10}{2}\bigg) = 7 \]

This leads to \[ 2 \times \frac{7}{2} = 7\]

Multiplying gives \[ 7 = 7 \], verifying the solution is correct.

This step is critical to ensure the solution is valid.

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Most popular questions from this chapter

Geoff raised \(\$ 65\) for cancer research for every mile he walked in a recent fund-raiser, in addition to the \(\$ 100\) he was donating himself. Write an equation for the relationship between the number of miles Geoff walked, \(m,\) and total amount he raised, \(r .\)

Solve this system of quadratic equations by drawing a graph. $$ y=x^{2} \quad y=4-3 x^{2} $$

When you draw a graph, you have to decide the range of values to show on each axis. Each exercise below gives an equation and a range of values for the \(x\) -axis. Use an inequality to describe the range of values you would show on the \(y\) -axis, and explain how you decided. (It may help to try drawing the graphs.) $$y=x^{2}+1 \text { when }-5 \leq x \leq 5$$

Examine this table. a. Use the table to estimate the solutions of \(t(t-3)=5\) to the nearest integer. b. If you were searching for solutions by making a table with a calculator, what would you have to do to find solutions to the nearest tenth? c. Find two solutions of \(t(t-3)=5\) to the nearest tenth. $$ \begin{array}{rr}{t} & {t(t-3)} \\ {-2} & {10} \\ {-1} & {4} \\ {0} & {0} \\\ {1} & {-2} \\ {2} & {-2} \\ {3} & {0} \\ {4} & {4} \\ {5} & {10} \\\ {6} & {18} \\ {7} & {28}\end{array} $$

For each table, tell whether the relationship between x and y could be linear, quadratic, or an inverse variation, and write an equation for the relationship. $$ \begin{array}{|c|cc|c|c|c|}\hline x & {0.5} & {2} & {4} & {5} & {20} \\\ \hline y & {-1.25} & {2.5} & {7.5} & {10} & {47.5} \\ \hline\end{array} $$
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