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Chapter 0: Problem 1

Solve for the indicated variable. $$9 m-7=11$$

Short Answer

Expert verified
The solution is \(m = 2\).

Step by step solution

01

Identify the Equation

The equation to solve is \(9m - 7 = 11\). Our goal is to isolate \(m\).
02

Eliminate the Constant Term

We need to eliminate the constant \(-7\) from the left side. To do this, add 7 to both sides of the equation. \[9m - 7 + 7 = 11 + 7\] This simplifies to: \[9m = 18\]
03

Solve for 'm'

Divide both sides of the equation by 9 to solve for \(m\). \[\frac{9m}{9} = \frac{18}{9}\]This simplifies to: \[m = 2\]

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Key Concepts

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

Solve for a Variable
Solving for a variable means finding the value of that variable in an equation. It is a crucial skill in algebra that allows you to determine what number a variable stands for. In the given exercise, we focused on finding the value of the variable \(m\) in the equation \(9m - 7 = 11\). To solve for \(m\), you aim to get \(m\) by itself on one side of the equation. A few key points to remember:
  • Identify what you're solving for. Here, the goal is to determine the value of \(m\).
  • Work systematically through each side of the equation until the variable stands alone.
  • Keep your equation balanced by performing the same operations on both sides.
Understanding these steps is the foundation for solving any algebraic equation.
Isolating Variables
Isolating a variable involves simplifying an equation so that the variable you鈥檙e solving for is alone on one side of the equation. This process typically requires applying inverse operations to each part of the equation. In our example, we first needed to get rid of the constant \-7\ on the left by adding 7 to both sides. Here's the breakdown:
  • Identify the operations affecting the variable. For \(9m - 7\), the \(-7\) needs to be eliminated first.
  • Apply the inverse operation to both sides. The inverse of subtraction is addition, so add 7 to both sides to cancel out the \(-7\).
  • Simplify the resulting expression. You should then be left with the variable term isolated, like \(9m = 18\).
This systematic isolation is crucial, especially when dealing with more complex equations.
Simplifying Equations
Simplifying equations is about reducing them to their simplest form. This often involves operations such as combining like terms, removing coefficients, or eliminating fractions. In our problem, the expression \(9m = 18\) was simplified further by dividing both sides by 9 to solve for \(m\). Key steps include:
  • Simplify the expressions on both sides of the equation as much as you can.
  • Apply the necessary operations to make the variable independent by using its coefficients.
  • Perform any arithmetic needed (like division or multiplication) to completely simplify the variable's side of the equation.
The result is a cleaner, more direct equation, like \(m = 2\), that is easy to interpret. Simplification helps maintain clarity as you solve for variables in different equations.

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