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Chapter 12: Problem 33

Solve the equation. Check your solution. $$0.3 x-\frac{3}{5} x+1.6=1.555$$

Short Answer

Expert verified
The solution to the equation \(0.3x - \frac{3}{5}x + 1.6 = 1.555\) is \(x = 0.15\). After substitution of 'x', both sides of the equation are equal confirming the solution is correct.

Step by step solution

01

Simplifying the Equation

Combine the like terms together. That means, the terms with the variable 'x' should be added together and moved to one side of the equation. \(0.3 x - \frac{3}{5}x = 1.555 - 1.6 \)
02

Solving for 'x'

Transform fractions into decimals for ease of calculation. The equation now will be: \(0.3x - 0.6x = -0.045\). Adding the terms on the left side will result in \(-0.3x = -0.045\), and further isolating 'x' you have \(x = \frac{-0.045}{-0.3}\)
03

Checking the Solution

Substitute the found value of 'x' back into the original equation: \(0.3x - \frac{3}{5}x + 1.6 = 1.555\). If both sides of the equation are equal, this solution is correct.

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Key Concepts

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

Combining Like Terms
When you encounter a linear equation like the one we have here, the first step is to simplify by combining like terms. Like terms are terms that have the exact same variable raised to the same power. In our equation, these terms are the ones with the variable \(x\): \(0.3x\) and \(-\frac{3}{5}x\). Our goal is to make the equation simpler by combining these.
  • This involves performing addition or subtraction on the coefficients (the numbers in front of the variables).
  • In this case, convert the fraction into a decimal to make calculations easier. \( -\frac{3}{5} \) is equivalent to \(-0.6\).
  • Now, combine \(0.3x\) and \(-0.6x\) to get \(-0.3x\).
After combining these like terms, the equation becomes much simpler and easier to solve!
Decimal and Fraction Conversion
Converting between decimals and fractions is a handy skill, especially when solving equations. It can help simplify calculations and make solving equations more straightforward. In our exercise, the conversion happens for \(-\frac{3}{5}\). Here's how you can deal with such conversions:
  • To convert a fraction to a decimal, divide the numerator (the top number) by the denominator (the bottom number). For \(-\frac{3}{5}\), this gives \(-0.6\).
  • This conversion is helpful because it aligns all the terms to the same type (i.e., all in decimal form), facilitating straightforward arithmetic operations.
  • Always double-check your conversions to avoid mistakes in calculations later in the equation-solving process.
By converting terms properly, you'll find that dealing with a mix of decimals and fractions becomes much less daunting and easier to manage!
Equation Checking
Checking your solution is a crucial step in solving any equation. It ensures that the solution found truly satisfies the equation. Here's how you can effectively check your solution:
  • Take the value of \(x\) you obtained and plug it back into the original equation.
  • For our equation \(0.3x - \frac{3}{5}x + 1.6 = 1.555\), substitute back the calculated \(x\) value.
  • Calculate both sides of the equation to see if they match. If they do, then your solution is correct! If not, you may need to recheck your steps.
This step acts as a final verification that helps catch mistakes, ensuring your solution is accurate and reliable! It's like your mathematical proof before wrapping up the problem.

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Most popular questions from this chapter

PROBLEM SOLVING You play a game that involves drawing two numbers from a hat. There are 25 pieces of paper numbered from 1 to 25 in the hat. Each number is replaced after it is drawn. Find the probability that you will draw the 3 on your first draw and a number greater than 10 on your second draw.

MAKING AN ARGUMENT A meteorologist claims that there is a 70\(\%\) chance of rain. When it rains, there is a 75\(\%\) chance that your softball game will be rescheduled. Your friend believes the game is more likely to be rescheduled than played. Is your friend correct? Explain your reasoning.

ERROR ANALYSIS Events \(A\) and \(B\) are independent. Describe and correct the error in finding \(P(A \text { and } B)\) . $$\begin{array}{l}{P(A)=0.6 \quad P(B)=0.2} \\ {P(A \text { and } B)=0.6+0.2=0.8}\end{array}$$

You and your friend are among several candidates running for class president. You estimate that there is a 45\(\%\) chance you will win and a 25\(\%\) chance your friend will win. What is the probability that you or your friend win the election?

ERROR ANALYSIS A shelf contains 3 fashion magazines and 4 health magazines. You randomly choose one to read, set it aside, and randomly choose another for your friend to read. Describe and correct the error in finding the probability that both events \(A\) and \(B\) occur. Event \(A :\) The first magazine is fashion. Event \(B\) : The second magazine is health. $$\begin{array}{ll}{P(A)=\frac{3}{7}}{P(B | A)=\frac{4}{7}} \\ {P(A \text { and } B)=\frac{3}{7} \cdot \frac{4}{7}=\frac{12}{49}} & {=0.245}\end{array}$$
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