/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 110 Solve each equation. $$ 0.10... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 110

Solve each equation. $$ 0.10(7)+1.00 x=0.30(7+x) $$

Short Answer

Expert verified
x = 2

Step by step solution

01

- Expand the parentheses

Expand both sides of the equation to remove the parentheses, yielding their respective products:\[ 0.10(7) + 1.00 x = 0.30(7 + x) \]\[ 0.7 + x = 2.1 + 0.30x \]
02

- Move variables to one side

Subtract \(0.30x\) from both sides of the equation to get all variable terms on one side:\[ 0.7 + x - 0.30x = 2.1 \]This simplifies to: \[ 0.7 + 0.70x = 2.1 \]
03

- Simplify the equation

Combine the like terms to get a simpler equation:\[ 0.7 + 0.70x = 2.1 \]
04

- Isolate the variable x

Subtract 0.7 from both sides to isolate the variable \(x\):\[ 0.70x = 1.4 \]
05

- Solve for x

Divide both sides of the equation by 0.70 to find the value of \(x\):\[ x = \frac{1.4}{0.70} \]Thus:\[ x = 2 \]

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Key Concepts

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

Expanding Parentheses
First, understand why expanding parentheses is important. This process helps to eliminate any grouping and makes solving the equation easier.
Let's take the equation: \(0.10(7) + 1.00x = 0.30(7 + x)\).
To expand the parentheses, distribute each term inside the parentheses by the number outside. Here's how:
  • \(0.10 \times 7 = 0.7\)
  • \(0.30(7) + 0.30(x) = 2.1 + 0.30x\)
After expanding, the equation becomes: \(0.7 + x = 2.1 + 0.30x\). Now, you have an equation without parentheses, ready for the next steps.
Isolating Variables
The goal of isolating variables is to get all the terms with the variable on one side of the equation. This lets you solve for that variable. In our example, after expanding, the equation is: \(0.7 + x = 2.1 + 0.30x\).
Next, we'll subtract \(0.30x\) from both sides to move \(x\) terms together: \(0.7 + x - 0.30x = 2.1\).
This simplifies to: \(0.7 + 0.70x = 2.1\). Isolating the variable step is essential because it helps in simplifying the equation step-by-step and reduces errors.
Combining Like Terms
Combining like terms involves adding or subtracting terms that have the same variable parts. After isolating the variable, the equation is: \(0.7 + 0.70x = 2.1\).
Here, the terms \(0.7\) and \(0.70x\) are on one side. To isolate \(x\), subtract \(0.7\) from both sides: \(0.70x = 1.4\).
Finally, solve for \(x\) by dividing both sides by 0.70: \(x = \frac{1.4}{0.7} = 2\). Combining like terms simplifies equations and helps in moving closer to the final solution efficiently.

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