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Chapter 2: Problem 107

Solve each equation, if possible. $$ -(4-m)=-10 $$

Short Answer

Expert verified
The solution to the equation is \( m = -6 \).

Step by step solution

01

Eliminate negative sign

We start by eliminating the negative sign from the left side of the equation. The given equation is \[- (4 - m) = -10. \]To remove the negative sign, we multiply the entire equation by -1 to obtain:\[ 4 - m = 10. \]
02

Isolate the variable

To isolate the variable \( m \), we need to get rid of the constant term \( 4 \) on the left side of the equation. We can do this by subtracting \( 4 \) from both sides of the equation:\[ 4 - m - 4 = 10 - 4. \]This simplifies to:\[ - m = 6. \]
03

Solve for the variable

The last step is to solve for \( m \) by multiplying or dividing both sides by \( -1 \), thus:\[ m = -6. \]

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Key Concepts

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

Solving Equations
Equation solving is a fundamental part of algebra. An equation, in simple terms, is a mathematical statement that shows that two expressions are equal. The goal is to find the values of the variables that make the equation true. The given exercise, \[-(4-m) = -10,\]is an example of a linear equation with one variable, and our task is to find the value of \(m\).
The solution process typically involves:
  • Removing signs to simplify the expression.
  • Isolating variables by performing operations like addition or subtraction on both sides.
  • Solving for the variable to find its value.
Solving equations often requires performing equivalent operations on both sides to maintain balance. Remember, whatever you do to one side of the equation, you must do to the other to keep it equal.
Variables and Constants
Understanding the differences between variables and constants is crucial when solving equations. A variable is a symbol, often represented by letters such as \(m\), \(x\), or \(y\), that stands in for an unknown number we want to find. In our exercise, \(m\) is the variable.
On the other hand, a constant is a fixed value that doesn't change. In the equation we are working with, \[-(4-m) = -10,\]the numbers \(4\) and \(-10\) are constants. Here is how you work with them:
  • Constants remain unchanged throughout the equation.
  • Variables need to be isolated to one side to solve the equation.
  • Use inverse operations to break down equations and isolate variables.
Distinguishing between constants and variables will help you solve equations more easily, allowing for clarity in each step.
Negative Numbers
Negative numbers are simply numbers less than zero and are usually represented with a \(-\) sign. They follow specific rules when performing mathematical operations.In solving the equation \[-(4-m) = -10,\]negative numbers appear in terms of signs. Understanding how to handle these is essential as follows:
  • Multiplying or dividing two negative numbers results in a positive number.
  • Adding a negative number is equivalent to subtracting its absolute value.
  • Subtracting a negative number can be rewritten as adding its absolute value.
In our solution, multiplying through the equation by \(-1\) helped to eliminate the negative signs, which simplified the equation considerably. Remember, proper handling of negative numbers is key to finding correct solutions in algebra.

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Most popular questions from this chapter

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