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Chapter 2: Problem 24

For exercises 23-54, (a) clear the fractions and solve. (b) check. $$ \frac{3}{5} x+2=17 $$

Short Answer

Expert verified
x = 25

Step by step solution

01

Eliminate the Fraction

Multiply every term by 5 to clear the fraction. The equation is currently \(\frac{3}{5} x + 2 = 17\). Multiplying each term by 5 results in the equation \(3x + 10 = 85\).
02

Isolate the Variable

Subtract 10 from both sides to isolate the variable term on one side. This gives \(3x = 75\).
03

Solve for x

Divide both sides by 3 to solve for \(x\). Thus, \(x = 25\).
04

Check the Solution

Substitute \(x = 25\) back into the original equation to verify the solution. The original equation is \(\frac{3}{5} x + 2 = 17\). Substitute 25 for \(x\): \(\frac{3}{5} \times 25 + 2 = 17\), which simplifies to \(15 + 2 = 17\). Since both sides are equal, the solution \(x = 25\) is correct.

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Key Concepts

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

fraction elimination
When solving equations involving fractions, the first step is often to eliminate the fractions for simplicity. This process is called 'fraction elimination'. If you see an equation like \(\frac{3}{5} x + 2 = 17\), your goal is to remove the fraction from the equation.
This can be achieved by multiplying every term of the equation by the common denominator of all the fractions involved. Here, the denominator is 5.
Multiplying \(\frac{3}{5} x + 2 = 17\) by 5, we get \(5 \times \frac{3}{5} x + 5 \times 2 = 5 \times 17\), resulting in \(3x + 10 = 85\).
Now, we have an equation without fractions, making it easier to solve.
isolating variables
After eliminating fractions, the next step is to isolate the variable. Our simplified equation is now \(3x + 10 = 85\).
To isolate the variable, move all other terms to the opposite side of the equation. Start by subtracting 10 from both sides, resulting in \(3x = 75\).
This step ensures that the variable term \(3x\) is by itself on one side of the equation, allowing you to then solve for x.
solution verification
Once you have found a possible solution, it鈥檚 crucial to verify it by substituting it back into the original equation. In our case, we found \(x = 25\).
Substitute \(x = 25\) into the original equation \(\frac{3}{5} x + 2 = 17\): we get \(\frac{3}{5} \times 25 + 2 = 17\).
Calculate \(\frac{3}{5} \times 25\), which equals 15, and then add 2, resulting in 17. Both sides of the equation are equal, confirming that \(x = 25\) is indeed the correct solution.
elementary algebra
These process steps are fundamental concepts in elementary algebra. They include tactics such as 'fraction elimination', 'isolating variables', and 'solution verification'.
Understanding these methods helps you handle more complicated equations. Practicing these steps ensures that dealing with variables and fractions in algebra becomes intuitive.
In problems where equations involve fractions, always remember that simplifying by eliminating fractions first makes the rest of the solving process much easier.

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Most popular questions from this chapter

In the first five meets of the season, a collegiate gymnast had vault scores of \(9.625,9.750,9.825,9.725\) and \(9.550\). Find the score she needs in the next meet to have an average score of at least 9.700.

For exercises 93-96, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve: \(-2 x+12>36\) Incorrect Answer: \(-2 x+12>36\) $$ \begin{aligned} -12 &-12 \\ \hline-2 x+0 &>24 \\ \frac{-2 x}{-2} &>\frac{24}{-2} \\ x &>-12 \end{aligned} $$

For problems 97-100, the symbol > means "greater than," and the symbol \(<\) means "less than." Replace the blank with < or \(>\) to create a true statement. $$ -12 $$_____0

For exercises 89-92, solve. Use a calculator to do the arithmetic. $$ -\frac{3}{8}=\frac{b}{160} $$

For exercises 1-8, use a number line graph to represent the inequality. $$ x<8 $$
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