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Chapter 6: Problem 10

Solve each equation. $$\frac{x}{2}-4=38$$

Short Answer

Expert verified
The solution to the equation is \(x = 84\).

Step by step solution

01

Remove -4 from the left side

In order to isolate \(x\) on one side of the equation, first add 4 on both sides. The equation becomes \(\frac{x}{2} - 4 + 4 = 38 + 4\), which simplifies to \(\frac{x}{2} = 42\).
02

Multiply both sides by 2

Now multiply both sides of the equation by 2 to remove the denominator of \(x\). The equation becomes \(2 * \frac{x}{2} = 42 * 2\), which simplifies to \(x = 84\).

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Key Concepts

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

Algebraic Manipulation
Algebraic manipulation is a fundamental skill in solving equations. It involves rearranging and simplifying expressions using arithmetic operations. Think of it like untangling a string鈥攊t becomes clearer as we separate parts of an equation. For instance, in the equation \(\frac{x}{2} - 4 = 38\), we aim to clear fractions or variables to solve for \(x\).

Here are common techniques used in algebraic manipulation:
  • Adding or subtracting the same number from both sides of the equation. This helps clear constants that are not associated with the variable.
  • Multiplying or dividing both sides by the same nonzero number. It's crucial in managing fractions or coefficients.
  • Combining like terms to simplify expressions further.
When applied correctly, these manipulations maintain equality while making the equation simpler to solve.
Isolating the Variable
Isolating the variable means getting the unknown (often represented as \(x\)) alone on one side of the equation. This is the goal of many algebra problems: find the value of the variable. Consider the equation \(\frac{x}{2} - 4 = 38\). We want \(x\) to be by itself without any fractions or other numbers nearby.

Follow these steps to isolate \(x\):
  • First, remove constant terms that are added or subtracted from the variable. Here, we added 4 to both sides to cancel out the \(-4\). This clears one step of the isolation process.
  • Next, address any fractions or coefficients. We multiplied both sides by 2 to eliminate the fraction, which gives \(x = 84\).
These actions are critical. They transform the equation into a simple solution. Once isolated, the variable's value becomes evident.
Equation Solving Techniques
There are various techniques used to solve an equation, each depending on its structure. Some include operations that help both isolate and simplify equations. Let's explore some basic yet effective techniques:

  • **Balancing both sides:** Whatever operation you apply to one side, also apply it to the other. This principle keeps the equation balanced鈥攖he essence of equality.
  • **Clearing fractions and decimals:** Fractions can complicate equations, so consider multiplying through by the denominator early on to simplify.
  • **Successfully substituting or transposing terms:** When equations get complex, restructuring them by changing positions of terms might be necessary.
By applying these techniques methodically, you gain greater control over the equation. You will be better equipped to understand and tackle complex problems. Every equation, simple or complex, boils down to these fundamental actions.

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Most popular questions from this chapter

Solve each equation. $$\frac{1}{a}+\frac{1}{b}=1 \text { for } a$$

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{x z-2 x}{y z-2 y}$$

Comparative shopping A 30-pound bag of fertilizer costs \(\$ 12.25,\) and an 80-pound bag costs \(\$ 30.25 .\) Which is the better buy?

$$\text { Solve: } x^{-2}+x^{-1}=0$$.

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{45}{9 a}$$
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