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Chapter 3: Problem 299

In the following exercises, solve for the unknown. $$ r-(-5)=-9 $$

Short Answer

Expert verified
r = -14

Step by step solution

01

Simplify the equation

Start by simplifying the left-hand side of the equation. Notice that subtracting a negative number is the same as adding the absolute value of that number. Thus, transform the expression: \[ r - (-5) = r + 5 \]
02

Rewrite the equation

After simplifying, rewrite the equation as: \[ r + 5 = -9 \]
03

Isolate the variable

To solve for \( r \), subtract 5 from both sides of the equation: \[ r + 5 - 5 = -9 - 5 \]
04

Simplify the right-hand side

Simplify the right-hand side by performing the subtraction: \[ r = -14 \]

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Key Concepts

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

subtracting negative numbers
When you see an equation like \( r - (-5) = -9 \), it might look confusing at first. The concept of subtracting a negative number can be tricky.

But here鈥檚 a simple trick: subtracting a negative is the same as adding its positive counterpart. So, \( r - (-5) \) can be thought of as \( r + 5 \).

This makes the equation easier to manage. You simply rewrite it:
  • Original: \( r - (-5) = -9 \)
  • Rewrite: \( r + 5 = -9 \)
Understanding this idea helps because it turns a seemingly complex term into something more familiar.

isolating the variable
Next, we want to solve for the unknown variable, which is \( r \) in this case. To do this, we need to get \( r \) by itself on one side of the equation.

We started with our simplified equation:
  • \( r + 5 = -9 \)
To isolate \( r \), we need to make sure it's the only term on one side of the equation. Right now, it鈥檚 being added to 5. So, we do the opposite operation to both sides: subtract 5.

  • \( r + 5 - 5 = -9 - 5 \)
This operation cancels out the +5 on the left side:

  • \( r = -14 \)
By isolating the variable \( r \), we have now found its value.
simplifying equations
Simplifying equations is an important step in solving them. Simplification helps ensure that the equation is as straightforward as possible.

In our example, we initially had:
  • \( r - (-5) = -9 \)
Through our understanding of subtracting negative numbers, we simplified it to:
  • \( r + 5 = -9 \)
Once the equation was simplified, isolating the variable became much easier. This shows why simplification is so crucial. You turn complex expressions into more manageable ones and can see the path to the solution more clearly.

Always simplify step-by-step:
  • Address tricky terms (like subtracting negatives)
  • Combine like terms
  • Perform straightforward arithmetic
This approach ensures you can solve equations efficiently.

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Most popular questions from this chapter

Simplify each absolute value expression. (a) \(|7|\) (b) \(|-25|\) (c) \(|0|\)

In the following exercises, translate and solve. Nine more than m is equal to 5.

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In the following exercises, divide. $$-180 \div 15$$

In the following exercises, simplify. $$ -15+(-12) $$
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