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Chapter 6: Problem 26

Solve each equation. $$ 7+\frac{6}{a}=5 $$

Short Answer

Expert verified
The solution is \(a = -3\).

Step by step solution

01

Isolate the fraction

To solve the equation \(7 + \frac{6}{a} = 5\), first isolate the fraction \(\frac{6}{a}\) by subtracting 7 from both sides. This will give us \(\frac{6}{a} = 5 - 7\).
02

Simplify the equation

Simplify the right side of the equation: \(5 - 7 = -2\). So the equation becomes \(\frac{6}{a} = -2\).
03

Solve for \(a\)

To solve for \(a\), multiply both sides by \(a\) to clear the fraction: \(6 = -2a\). Then, divide both sides by \(-2\) to solve for \(a\): \(a = \frac{6}{-2}\).
04

Simplify the solution

Simplify the fraction \(\frac{6}{-2}\): \(\frac{6}{-2} = -3\). Thus, \(a = -3\).

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Key Concepts

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

Equation Solving
One of the first steps in solving an equation is to identify what you are trying to find. In this problem, you want to determine the value of the variable \(a\). Begin by expressing the equation clearly: \(7 + \frac{6}{a} = 5\).

**Step-by-Step Approach:**
  • Start by isolating any fractions or additional terms on one side of the equation. This is key to simplifying the equation later on.
  • Simplify the equation by collecting like terms or removing constants from both sides of the equation.


For instance, subtracting 7 from both sides of our given equation allows you to isolate the fractional component, leading to \(\frac{6}{a} = -2\). This process transforms the equation so you can focus on simplifying and solving it further.
Algebraic Manipulation
Algebraic manipulation is a crucial skill in mathematics. It involves rearranging and simplifying equations to isolate variables. In our example, once you have \(\frac{6}{a} = -2\), the next step is to clear the fraction for easier manipulation.

**Simplify by Eliminating Fractions:**
  • Multiply both sides by the denominator to eliminate fractions. In this case, multiply both sides by \(a\), resulting in \(6 = -2a\).
  • Once the fraction is cleared, work on isolating \(a\) by dividing both sides of the equation by the coefficient of \(a\).


It is important to handle each operation carefully to maintain the equation's equality. This approach is essential when solving equations with single variables.
Negative Numbers
Understanding and working with negative numbers is fundamental in algebra. In the solution of the equation \(6 = -2a\), you deal with negatives when dividing both sides by \(-2\).

**Key Points about Negatives:**
  • A negative divided by a negative results in a positive number. This is why \(a = \frac{6}{-2}\) simplifies to \(a = -3\).
  • Pay attention to signs during all algebraic manipulations to avoid errors. Misplacing a negative sign can lead to an incorrect solution.


By recognizing and managing negative numbers properly, you can confidently solve equations and simplify complex algebraic expressions.

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Most popular questions from this chapter

Find each square root. See Section 1.3. $$ \sqrt{4} $$

A board of length \(\left(3 x^{4}+6 x^{2}-18\right)\) meters is to be cut into three pieces of the same length. Find the length of each piece.

If the area of a parallelogram is \(\left(2 x^{2}-17 x+35\right)\) square centimeters and its base is \((2 x-7)\) centimeters, find its height.

Factor the following. $$ 2 x^{3}-32 x $$

If the volume of a box is \(\left(x^{4}+6 x^{3}-7 x^{2}\right)\) cubic meters, its height is \(x^{2}\) meters, and its length is \((x+7)\) meters, find its width.
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