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Chapter 5: Problem 305

In the following exercises, solve the equation. $$ 0.5 x=3.5 $$

Short Answer

Expert verified
x=7

Step by step solution

01

Write the Equation

Start with the given equation: 0.5x = 3.5
02

Isolate the Variable

To solve for x, divide both sides of the equation by 0.5: $$\frac{0.5x}{0.5}=\frac{3.5}{0.5}$$
03

Simplify

Simplify both sides of the equation: $$x=7$$

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Key Concepts

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

Isolating the Variable
The ultimate goal when solving linear equations is to find the value of the unknown variable. This process begins with isolating the variable. Imagine the equation like a balanced scale, with the variable on one side and the numbers on the other.
To isolate the variable, you need to get it by itself on one side of the equation.
This often involves performing inverse operations, those that 'undo' other operations.
For example, if the variable is being multiplied by a number, you would divide both sides of the equation by that number to cancel it out. This keeps the equation balanced while making it easier to solve for the unknown.
Division in Equations
Division is a common method to isolate the variable in an equation. Let's use the example from the original exercise: \(0.5x = 3.5\). Here, the variable \(x\) is being multiplied by 0.5.
To isolate \(x\), you need to divide both sides of the equation by 0.5.
This is written as:
  • \(\frac{0.5x}{0.5} = \frac{3.5}{0.5}\)
  • \(x = 7\)
This process cancels out the 0.5 on the left side, leaving you with just \(x\).
Remember to always perform the same operation on both sides of the equation to maintain balance.
Simplification of Equations
Simplifying an equation means making it as straightforward as possible. After isolating the variable and performing any necessary operations, check if the equation can be simplified further. For example, in the final step of our exercise: \(x = 7\), the equation is already as simple as it can be.
If you had fractions or decimals, you might need to perform additional operations to simplify the result.
Simplification ensures the solution is clear and concise.
Breaking down the equation into smaller, easily manageable steps can also make complex problems much easier to solve.
This way, you're not just finding the answer but also understanding the process!

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