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Chapter 11: Problem 1

Solve. Clear fractions first. $$ \frac{1}{2} x+3=5 $$

Short Answer

Expert verified
x = 4

Step by step solution

01

Identify the Fraction

Notice that the equation \(\frac{1}{2} x + 3 = 5\) involves a fraction, \(\frac{1}{2}\). The goal is to clear this fraction first.
02

Multiply by the Denominator

Multiply both sides of the equation by 2 (the denominator of the fraction) to eliminate the fraction: \(\frac{1}{2} x \cdot 2 + 3 \cdot 2 = 5 \cdot 2\). This simplifies to \(x + 6 = 10\).
03

Isolate the Variable

Subtract 6 from both sides to isolate the variable \(x\): \(x + 6 - 6 = 10 - 6\), which simplifies to \(x = 4\).

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Key Concepts

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

Clearing Fractions
When solving linear equations, dealing with fractions can complicate things. Clearing fractions makes the equation simpler to work with.

To clear a fraction, you need to eliminate it by doing an operation that leaves you with whole numbers. The common method for clearing fractions involves:
  • Identifying the denominator of the fraction.
  • Multiplying every term in the equation by this denominator.
For example, in the equation \(\frac{1}{2} x + 3 = 5\), the fraction is \(\frac{1}{2}\). By multiplying every term by 2 (the denominator), you eliminate the fraction:

\[ \frac{1}{2} x \times 2 + 3 \times 2 = 5 \times 2 \] which simplifies to \[ x + 6 = 10. \]
This makes the equation easier to solve.
Isolating Variables
Isolating the variable is a crucial step in solving linear equations. It involves getting the variable by itself on one side of the equation. This lets you see the solution directly.

In the equation \( x + 6 = 10 \), our goal is to have \( x \) on one side alone. To do this, we need to remove the 6 from the left side. We do this by performing the opposite operation:
  • Since 6 is added to \( x \), we subtract 6 from both sides of the equation.
Write it out:
\[ x + 6 - 6 = 10 - 6 \] which simplifies to \( x = 4 \).

Now, the variable \( x \) is isolated. Isolating the variable helps find its value directly.
Multiplication in Equations
Understanding how to correctly use multiplication in equations is essential for manipulating and solving them.
Multiplication helps in two main scenarios:
  • Clearing fractions by using the multiplicative inverse (denominator).
  • Maintaining balance while performing operations on both sides.
In our example, we used multiplication to clear the fraction \(\frac{1}{2} x \):
\[ \frac{1}{2} x \times 2 + 3 \times 2 = 5 \times 2 \].

This step ensures that all terms are whole numbers, making the equation easier to manipulate.
Remember, whatever you multiply by on one side, you must also multiply by on the other side. This keeps the equation balanced and valid.

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