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Chapter 4: Problem 666

In the following exercises, solve the equation. $$ \frac{x}{5}=-10 $$

Short Answer

Expert verified
x = -50

Step by step solution

01

Identify the equation

The given equation is: \[ \frac{x}{5} = -10 \]
02

Isolate the variable

To solve for \( x \), multiply both sides of the equation by 5 to get rid of the fraction: \[ 5 \times \frac{x}{5} = 5 \times (-10) \]
03

Simplify the equation

Simplify both sides of the equation: \[ x = -50 \]

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Key Concepts

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

Isolating Variables
When solving linear equations, one of the first steps often involves isolating the variable.
The goal is to get the variable (usually represented by letters like \( x \) or \( y \)) by itself on one side of the equation.
To do this, you usually need to remove any numbers that are being added, subtracted, multiplied, or divided by the variable.
For instance, in the equation \( \frac{x}{5} = -10 \), the variable \( x \) is being divided by 5.
To isolate \( x \), you need to get rid of the 5 by performing an operation that will cancel the division.
Multiplication Property of Equality
The multiplication property of equality states that you can multiply both sides of an equation by the same number without changing the solution.
This is extremely helpful for isolating variables.
In our example \( \frac{x}{5} = -10 \), we want to eliminate the fraction.
To do this, we multiply both sides of the equation by 5:
< ul > \[\begin{align*} 5 \times \frac{x}{5} &= 5 \times (-10) \end{align*}\]
This multiplication helps to cancel out the 5 on the left side and simplifies the right side to \( -50 \).
Now the equation is simplified to \( x = -50 \).
Working with Fractions
Working with fractions can feel tricky, but it's more manageable when you understand how to isolate variables and use properties of equality.
For instance, in an equation like \( \frac{x}{5} = -10 \), the fraction shows that \( x \) is divided by 5.
To eliminate the fraction, you perform the opposite operation, which is multiplication.
By multiplying both sides of the equation by 5, you clear the fraction: \[\begin{align*}5 \times \frac{x}{5} = 5 \times (-10)\end{align*}\] The fraction cancels out because \( 5 \times \frac{1}{5} = 1 \):
This leaves you with \( x = -50 \), a much simpler equation to work with.

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Most popular questions from this chapter

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. 4\(p^{2} q\) when \(p=-\frac{1}{2}\) and \(q=\frac{5}{9}\)

In the following exercises, perform the indicated operation. $$ 6 \frac{2}{5}+7 \frac{3}{5} $$

Decorating Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs \(\frac{3}{16}\) yard of print fabric and \(\frac{3}{8}\) yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover?

In the following exercises, subtract. $$\frac{11}{12}-\frac{5}{12}$$

In the following exercises, perform the indicated operation. $$ 3 \frac{5}{8}+3 \frac{7}{8} $$
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