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Chapter 2: Problem 57

Solve using the multiplication principle. Don't forget to check! $$ \frac{-x}{6}=9 $$

Short Answer

Expert verified
x = -54

Step by step solution

01

Isolate the Variable

To solve for the variable, multiply both sides of the equation by 6 to eliminate the fraction. This gives us: \[ \frac{-x}{6} \times 6 = 9 \times 6 \] which simplifies to: \[ -x = 54 \]
02

Solve for the Variable

To solve for \( x \), multiply both sides by -1 to get the variable by itself: \[ -x \times (-1) = 54 \times (-1) \] This results in: \[ x = -54 \]
03

Check the Solution

Substitute \( x = -54 \) back into the original equation to check your work: \[ \frac{-(-54)}{6} = 9 \] Simplify the left side: \[ \frac{54}{6} = 9 \] Since both sides are equal, the solution is correct.

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Key Concepts

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

Solving Linear Equations
Linear equations are equations where the highest power of the variable is one. These equations can often look complicated at first, but breaking them down step-by-step can make them easier to solve.
When dealing with linear equations, you typically aim to isolate the variable. By performing inverse operations, you move all the variable terms to one side and the constant terms to the other. This process often involves adding, subtracting, multiplying, or dividing both sides of the equation to maintain equality.
In our example, the equation is \(\frac{-x}{6}=9\). The goal is to get \(-x\) by itself on one side of the equation so we can then solve for \x\.
Isolating Variables
Isolating the variable is a critical step when solving equations. This means getting the variable on one side of the equation and everything else on the other side.
For the given equation \(\frac{-x}{6}=9\), start by eliminating the denominator. Multiply both sides by 6 to get rid of the fraction:
\(\frac{-x}{6} \times 6 = 9 \times 6\).
Simplifying this, we get \-x = 54\.
At this point, we have \-x\ isolated, but to solve for \x\, multiply both sides by -1:
\-x \times (-1) = 54 \times (-1)\. This simplifies to \x = -54\.
This process ensures that we maintain the balance of the equation while getting the variable by itself.
Checking Solutions in Algebra
One of the most important steps in solving equations is checking your solution. This verifies that your answer is indeed correct by substituting it back into the original equation.
For our example, we found \x = -54\. To check this, substitute \-54\ back into the original equation:\(\frac{-(-54)}{6}=9\).
Simplify the left side: \-(-54)\ becomes \54\, and \(\frac{54}{6}\) equals 9.
Since both sides of the equation are equal, our solution \x = -54\ is correct.
Always perform this step to confirm your answer and to ensure there were no mistakes in the calculation process.

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Most popular questions from this chapter

To prepare for Section \(2.6,\) review inequalities. Write a true sentence using either \(<\) or \(>\) $$-4 \square 7$$

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