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Chapter 3: Problem 28

Solve and check each equation. $$-54=6 m$$

Short Answer

Expert verified
We are given the equation \(-54 = 6m\). To solve for m, we need to isolate it. We do this by dividing both sides of the equation by 6: \[\frac{-54}{6} = \frac{6m}{6}\] which simplifies to \[-9 = m\]. To check our solution, we plug the found value of m back into the original equation and confirm that both sides are equal: \(-54 = 6(-9)\), which simplifies to \(-54 = -54\). Thus, the correct answer is \(m = -9\).

Step by step solution

01

Identify the given equation

The given equation is: \(-54 = 6m\)
02

Isolate the variable (m)

To isolate the variable m, we need to divide both sides of the equation by 6: \[\frac{-54}{6} = \frac{6m}{6}\]
03

Solve for m

After dividing, we find the value of m: \[-9 = m\]
04

Check the solution

To check the solution, substitute the value of m back into the original equation: \(-54 = 6(-9)\) Evaluate the expression on the right-hand side: \(-54 = -54\) Since both sides of the equation are equal, our solution is correct. We have verified that \(m = -9\).

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Key Concepts

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

Solving Equations
When tackling any linear equation like \(-54 = 6m\), our goal is to find the value of the unknown that satisfies the equation. This process is known as solving an equation. Linear equations involve variables raised to the power of one, making them the most straightforward equations to solve.
A methodical approach usually involves:
  • Understanding the equation structure and operations involved.
  • Applying inverse operations to simplify and isolate the variable.
  • Maintaining the balance of the equation by performing operations on both sides.
By carefully performing these steps, we can unravel what value makes the statement true.
Algebraic Expressions
At the heart of equations lie algebraic expressions. An algebraic expression may include numbers, variables, and operations like addition or multiplication. In our exercise, \(6m\) is such an expression where 6 is the coefficient and \(m\) is the variable.
Algebraic expressions allow us to:
  • Represent complex relationships in a simple, compact form.
  • Perform operations within them based on algebraic rules.
  • Transform and manipulate them to facilitate the solution process.
Understanding expressions allows us to effectively communicate mathematical concepts and solve equations accurately.
Variable Isolation
Isolating the variable is a critical step in solving equations. This involves rearranging the equation so that the variable stands alone on one side, ideally with a coefficient of one. In the equation \( -54 = 6m \), isolating \(m\) requires dividing both sides by 6.
The steps are:
  • Identify the operation involving the variable. Here, it is multiplication by 6.
  • Use the inverse operation, which is division, to isolate the variable.
  • Apply this operation on both sides to maintain balance.
Successfully isolating the variable brings us to the solution \( m = -9 \). Variable isolation transforms the problem into a simpler form and makes understanding and verifying the solution easier.

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