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Chapter 2: Problem 22

Solve each equation. See Examples 3 through \(5 .\) $$ 0.40 x+0.06(30)=9.8 $$

Short Answer

Expert verified
The solution to the equation is \( x = 20 \).

Step by step solution

01

Expand the equation

First, distribute the 0.06 to the 30 inside the parentheses. This will simplify the equation. The equation becomes: \[ 0.40x + 0.06 \times 30 = 9.8 \] Calculate the multiplication: \[ 0.06 \times 30 = 1.8\] So the equation transforms into: \[ 0.40x + 1.8 = 9.8 \]
02

Isolate the variable term

Subtract 1.8 from both sides of the equation to get the term with the variable alone on one side. \[ 0.40x + 1.8 - 1.8 = 9.8 - 1.8 \] This simplifies to: \[ 0.40x = 8.0 \]
03

Solve for the variable

Divide both sides of the equation by 0.40 to solve for \( x \). \[ \frac{0.40x}{0.40} = \frac{8.0}{0.40} \] Calculate the division on the right side: \[ x = 20 \]

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Key Concepts

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

Distributive Property
In the world of algebra, the distributive property is like a handy toolkit for simplifying expressions and equations. When you have an equation with a number outside parentheses, like in our exercise, you use the distributive property to "distribute" that number to everything inside the parentheses.

Here's how it works:
  • Take the number outside the parentheses (in our case, 0.06) and multiply it by each term inside the parentheses (30).
  • This math process is: \[0.06 \times 30 = 1.8\]
  • The equation then changes from \[0.40x + 0.06(30) = 9.8\]to \[0.40x + 1.8 = 9.8\]
Using the distributive property effectively simplifies the equation, allowing us to then focus on just the variable term and constants outside the parentheses. This is the first step in our journey to solve the equation.
Isolating the Variable
Once we've used the distributive property, our main goal becomes isolating the term with the variable. This is an essential skill in solving linear equations because it's all about transforming the equation so that the variable stands on its own on one side of the equation.

Here's the process:
  • You want the variable term, in this case, \(0.40x\), to be the only term on one side of the equation.
  • We had \[0.40x + 1.8 = 9.8\]
  • Subtract 1.8 from both sides to remove it from the left side:\[0.40x + 1.8 - 1.8 = 9.8 - 1.8\]
  • This simplifies to:\[0.40x = 8.0\]
By subtracting 1.8 from both sides, we have successfully isolated \(0.40x\), preparing us to easily solve for the variable in the next step. Remember to always perform the same operation on both sides to maintain the balance of the equation.
Solving Equations
The final mission is to solve the equation, which means finding the exact value of the variable. Now that \(0.40x\) is isolated, the final step is straightforward.

The steps to solve for \(x\) are:
  • We have the equation \[0.40x = 8.0\]
  • We need to divide both sides by 0.40 to solve for \(x\): \[\frac{0.40x}{0.40} = \frac{8.0}{0.40}\]
  • This simplifies to:\[x = 20\]
And there we have it! The variable \(x\) equals 20. Always remember that whatever operation you use to isolate the variable, you must apply it equally to both sides of the equation. This keeps the equation balanced, leading you to the correct solution.

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Most popular questions from this chapter

The sum of \(\frac{1}{5}\) and twice a number is equal to \(\frac{4}{5}\) subtracted from three times the number. Find the number.

The product of twice a number and three is the same as the difference of five times the number and \(\frac{3}{4}\). Find the number.

The graph shows the number of U.S. Starbucks locations from 2002 to \(2008 .\) The height of the graph for each year shown corresponds to the number of Starbucks locations in the United States. Use this graph to answer . (We study graphs such as this further in Section 6.1.) Between which two years did the greatest increase in the number of Starbucks locations occur?

The number of counties in California and the number of counties in Montana are consecutive even integers whose sum is 114 . If California has more counties than Montana, how many counties does each state have? (Source: The World Almanac and Book of Facts)

Solve. For Exercises 1 through \(4,\) write each of the following as equations. The difference of three times a number and 1 is the same as twice the number. Find the number.
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