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Chapter 2: Problem 24

Write a two-column proof. $$\text { If } 4-\frac{1}{2} a=\frac{7}{2}-a, \text { then } a=-1$$

Short Answer

Expert verified
If you solve the equation, you'll find that \(a = -1\), and verifying it shows the solution is correct.

Step by step solution

01

Equate the given expressions

We start with the given equation: \[4 - \frac{1}{2}a = \frac{7}{2} - a.\]The goal is to manipulate this equation to isolate the variable \(a\).
02

Eliminate fractions by multiplying

To simplify the equation, first multiply every term by 2 to eliminate the fractions:\[2(4) - 2 \left(\frac{1}{2}a\right) = 2 \left(\frac{7}{2}\right) - 2(a),\] which simplifies to:\[8 - a = 7 - 2a.\]
03

Rearrange terms to isolate variables

Add \(2a\) to both sides of the equation to bring all terms involving \(a\) to one side:\[8 - a + 2a = 7 - 2a + 2a,\]which simplifies to:\[8 + a = 7.\]
04

Solve for the variable a

Subtract 8 from both sides to solve for \(a\):\[a = 7 - 8,\]which results in:\[a = -1.\]
05

Verify the solution

Substitute \(a = -1\) back into the original equation to verify:Original equation: \[4 - \frac{1}{2}(-1) = \frac{7}{2} - (-1).\]Simplify both sides:\[4 + \frac{1}{2} = \frac{7}{2} + 1.\]Converting to fractions:\[\frac{9}{2} = \frac{9}{2},\]which verifies the solution is correct.

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Key Concepts

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

Understanding Equations
An equation is a mathematical statement that asserts the equality between two expressions. In simple terms, it's like a balance scale where both sides must weigh the same. The given equation, \(4 - \frac{1}{2}a = \frac{7}{2} - a\), indicates this balance between the two sides of the equation.
The process of working with equations involves operations to simplify or rearrange terms:
  • Addition, subtraction, multiplication, and division can be used to both sides.
  • This ensures the balance of the equation is maintained.
  • Using these operations lets us solve for unknown variables, like \(a\) in this case.
In the problem, we initially have an equation with fractions. We need to first clear or manage these before proceeding to solve the equation further.
Isolate Variable
Isolating the variable means to get the variable on one side of the equation, all by itself. This is a key step in solving equations because once we isolate the variable, we can easily figure out its value. Here's how we do it:
  • To start, we have the equation: \(8 - a = 7 - 2a\).
  • We want all terms with \(a\) on one side. Add \(2a\) to both sides which gets us \(8 + a = 7\).
  • Now, to isolate \(a\), subtract 8 from both sides. This simplifies to \(a = 7 - 8\).
You end up with \(a = -1\), showing where the variable stands once everything else has been moved away.
This technique of isolating the variable is crucial as it gives us the clean answer directly. It's like solving a mystery step by step until you find out what the variable actually represents.
Verify Solution
Once you've found a solution to your equation, checking it is a good practice to ensure accuracy. Verifying a solution means plugging the value back into the original equation to see if it holds true:
  • Substitute \(a = -1\) back into \(4 - \frac{1}{2}(-1) = \frac{7}{2} - (-1)\).
  • Simplify both sides: left side becomes \(4 + \frac{1}{2}\), right side becomes \(\frac{7}{2} + 1\).
  • When converted to fractions, both sides equate to \(\frac{9}{2}\).
Since the calculations balance out, we confirm that \(a = -1\) is indeed the correct solution.
Verifying solutions prevents errors and gives confidence in your problem-solving skills. It's a great way to make sure that the answer isn't just plausible but actually correct.

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