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Chapter 2: Problem 56

For exercises 55-64, (a) clear the decimals and solve. (b) check. $$ 0.2 x+5=8 $$

Short Answer

Expert verified
x = 15.

Step by step solution

01

Clear the Decimals

To clear the decimals, multiply every term in the equation by 10 to get rid of the decimal point. The equation is: \(0.2x + 5 = 8\). Multiplying every term by 10 gives us: \(2x + 50 = 80\).
02

Solve for x

Now, solve the equation for \(x\). Start by subtracting 50 from both sides: \(2x + 50 - 50 = 80 - 50\). This simplifies to: \(2x = 30\). Next, divide both sides by 2: \(x = \frac{30}{2}\). This simplifies to: \(x = 15\).
03

Check the Solution

Substitute \(x = 15\) back into the original equation to verify the solution is correct. Original equation: \(0.2x + 5 = 8\). Substitute \(x = 15\): \(0.2(15) + 5 = 8\). Calculate \(0.2 \times 15\) to get 3, then: \(3 + 5 = 8\). Since both sides of the equation are equal, \(x = 15\) is the correct solution.

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Key Concepts

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

clearing decimals
When solving linear equations that have decimals, the first step is to 'clear' the decimals. This means converting the equation into one that doesn't have any decimal points, making it simpler to solve.
To clear decimals, find the smallest power of 10 that can turn all decimal numbers in the equation into whole numbers, and then multiply every term by this power of 10.

For example, in the equation: \(0.2x + 5 = 8\), the smallest power of 10 that works is 10 (since \(0.2\) becomes \(2\) when multiplied by 10).
Multiply every term by 10:
\(10 \times 0.2x + 10 \times 5 = 10 \times 8\)
This simplifies to:
\(2x + 50 = 80\).
Now, the equation is free of decimals, making it easier to solve.
multiplication to eliminate decimals
Multiplication is a powerful tool to eliminate decimals. As discussed in the previous section, you use multiplication to turn decimal numbers into whole numbers by choosing the right power of 10.

In the equation \(0.2x + 5 = 8\), multiplying by 10 helps get rid of the decimal in \(0.2\), transforming it into a simpler form: \(2x + 50 = 80\).

This method works because multiplying by 10 (or 100, or 1000, etc.) shifts the decimal point to the right, converting decimal numbers into whole numbers.
  • If your decimals have one decimal place, multiply by 10.
  • If your decimals have two decimal places, multiply by 100.
  • And so on...
Using this technique streamlines the equation and prepares it for further steps in solving, like isolating the variable.
isolating the variable
After clearing decimals, the next step in solving the equation is isolating the variable. This means getting the variable (in this case, \(x\)) by itself on one side of the equation.

Looking at our modified equation: \(2x + 50 = 80\), we need to perform operations that simplify the equation to solve for \(x\).
First, subtract 50 from both sides:
\(2x + 50 - 50 = 80 - 50\)
This simplifies to:
\(2x = 30\).

Next, divide both sides by 2 to isolate \(x\):
\(x = \frac{30}{2}\).
Solving this gives us:
\(x = 15\).

Breaking down each step:
  • Subtracting mimics the 'undoing' of the addition in the equation.
  • Dividing undoes the multiplication, leaving \(x\) by itself.
These steps help to isolate \(x\), yielding the solution to the equation.

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