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Chapter 4: Problem 63

Solve. If no equation is given, perform the indicated operation. $$ 17 x-25 x=\frac{1}{3} $$

Short Answer

Expert verified
\( x = -\frac{1}{24} \)

Step by step solution

01

Combine Like Terms

Begin by combining the terms on the left side of the equation. The terms are like terms since they both contain the variable \( x \). Combine \( 17x \) and \( -25x \) by subtracting the coefficients: \( 17 - 25 = -8 \). Thus, the equation simplifies to \( -8x = \frac{1}{3} \).
02

Solve for x

To solve for \( x \), divide both sides of the equation by \(-8\) to isolate \( x \). This gives: \[ x = \frac{\frac{1}{3}}{-8} = \frac{1}{3} \times \frac{1}{-8} = \frac{1}{-24} = -\frac{1}{24}. \] Thus, \( x = -\frac{1}{24} \).

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Key Concepts

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

Combining Like Terms
When solving linear equations, an essential step is combining like terms. Like terms are terms that have the same variable raised to the same power. In the given equation, both terms on the left-hand side, \(17x\) and \(-25x\), are like terms because they share the variable \(x\).

Combining these terms means simplifying them by adding or subtracting their coefficients. In this case:
  • The coefficient of \(17x\) is 17.
  • The coefficient of \(-25x\) is -25.
To combine, you perform the operation \(17 - 25\), which equals \(-8\). That's why \(17x - 25x\) simplifies to \(-8x\).

Simplifying like terms is crucial because it reduces the complexity of the equation, allowing you to focus on solving for the unknown easily.
Isolating the Variable
Once the equation has been simplified, the next step is to isolate the variable. "Isolating the variable" involves manipulating the equation so that the variable is alone on one side of the equation.

For the equation \(-8x = \frac{1}{3}\), the goal is to have \(x\) by itself. To do this, you'll divide both sides by \(-8\), the coefficient of \(x\).

Dividing by a negative number flips the sign, so it's essential to maintain balance by performing the same operation on both sides. Therefore, \(x = \frac{\frac{1}{3}}{-8}\).
  • This process is crucial because it isolates \(x\), making it easier to find the value of the variable.
Understanding this step is key when solving any equation; always aim to get the variable by itself with a coefficient of 1.
Fraction Division
Fraction division comes into play when isolating the variable on one side of the equation involves fractions.

Let's look at the step \(x = \frac{\frac{1}{3}}{-8}\). Dividing fractions might seem tricky at first, but it's essentially a process of multiplication. When dividing by a fraction, multiply by its reciprocal instead.
  • Reciprocal of \(-8\) is \(-\frac{1}{8}\).
Thus, to divide \(\frac{1}{3}\) by \(-8\), multiply \(\frac{1}{3}\) by \(-\frac{1}{8}\):

\[x = \frac{1}{3} \times -\frac{1}{8} = -\frac{1}{24}.\]

Fraction division simplifies multiplication and helps find the precise solution to such equations. Always remember: flipping and multiplying is a handy trick for dividing fractions efficiently.

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Most popular questions from this chapter

Solve. See the Concept Checks in this section. Round the mixed number \(7 \frac{1}{8}\) to the nearest whole number.

Perform each indicated operation. If the result is an improper fraction, also write the improper fraction as a mixed number. $$3+\frac{1}{2}$$

Answer true or false for each statement. It is possible for the average of two numbers to be less than both numbers.

A homeowner has \(15 \frac{2}{3}\) feet of plastic pipe. She cuts off a \(2 \frac{1}{2}\) -foot length and then a \(3 \frac{1}{4}\) -foot length. If she now needs a 10 -foot piece of pipe, will the remaining piece do? If not, by how much will the piece be short?

Recall that to find the average of two numbers, we find their sum and divide by \(2 .\) For example, the average of \(\frac{1}{2}\) and \(\frac{3}{4}\) is \(\frac{\frac{1}{2}+\frac{3}{4}}{2}\). Find the average of each pair of numbers. Study Exercise \(67 .\) Without calculating, \(\operatorname{can} \frac{1}{3}\) be the average of \(\frac{1}{2}\) and \(\frac{8}{9}\) ? Explain why or why not.
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