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Chapter 2: Problem 30

Solve and check. Label any contradictions or identities. $$-\frac{7 a}{8}-2=1$$

Short Answer

Expert verified
The solution is \(a = -\frac{24}{7}\).

Step by step solution

01

- Isolate the variable term

Add 2 to both sides of the equation to isolate the term containing the variable: display: \(-\frac{7a}{8} - 2 + 2 = 1 + 2\)
02

- Simplify the equation

Simplify the right side of the equation:\(-\frac{7a}{8} = 3\)
03

- Solve for the variable

Multiply both sides of the equation by the reciprocal of \(-\frac{7}{8}\) to solve for \(a\):\(a = 3 \times -\frac{8}{7}\)which simplifies to:\(a = -\frac{24}{7}\)
04

- Check the solution

Substitute \(a = -\frac{24}{7}\) back into the original equation to verify the solution:\(-\frac{7(-\frac{24}{7})}{8} - 2 = 1\)Simplify inside the parentheses:\(\frac{168}{56} - 2 = 1\)\(3 - 2 = 1\)Since both sides are equal, the solution is verified.

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Key Concepts

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

Isolating Variables
When solving linear equations, one of the primary steps is to isolate the variable we are solving for. Let's use the equation from the exercise as an example: \[-\frac{7 a}{8}-2=1\].
To isolate the variable term, we need to get rid of the added or subtracted constants on the same side as the variable.
Here鈥檚 how we do it:
  • Start by adding 2 to both sides of the equation to remove the -2 next to the fraction.
  • This results in: \[-\frac{7 a}{8} = 3\].
Now, the variable term is isolated on one side of the equation, making it easier to solve the next steps.
Reciprocal Multiplication
Once we have isolated the variable term, the next step is to solve for the variable by using reciprocal multiplication. In our example, we have: \[ -\frac{7 a}{8} = 3\].
To solve for \(a\), we multiply both sides by the reciprocal of \(-\frac{7}{8}\), which is \(-\frac{8}{7}\).
  • The reason we do this is to cancel out the fraction in front of the variable.
  • Thus, we perform the multiplication: \[ a = 3 \times -\frac{8}{7} \].
  • Simplifying this, we get: \[ a = -\frac{24}{7} \].
We now have the value of the variable.
Verifying Solutions
To ensure our solution is correct, we need to substitute the value of the variable back into the original equation and see if it holds true. For our problem:\[-\frac{7 a}{8} - 2 = 1\], substitute \( a = -\frac{24}{7}\).
Let's go through the steps:
  • Replace \( a \) with \( -\frac{24}{7} \): \[-\frac{7(-\frac{24}{7})}{8} - 2 = 1\].
  • Simplify inside the parentheses: \[-\frac{7(-\frac{24}{7})}{8} = \frac{168}{56} = 3\].
  • So, the equation becomes: \[ 3 - 2 = 1\].
  • Both sides are equal, verifying our solution.
Since the original equation is satisfied with \( a = -\frac{24}{7} \), we can be confident in our answer.

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Solve $$ 2(5-x)=\frac{1}{2}(x+1)[2.2] $$

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