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Chapter 1: Problem 19

Solve the given equations. $$2(s-4)=s$$

Short Answer

Expert verified
The solution to the equation is \(s = 8\).

Step by step solution

01

Distribute

The equation given is \(2(s-4)=s\). Begin by distributing the 2 into the terms inside the parentheses to eliminate the parentheses. This results in \(2s - 8 = s\).
02

Isolate Variable Terms

Next, try to get all terms containing the variable \(s\) on one side of the equation. Subtract \(s\) from both sides to keep the variable on the left side:\(2s - s - 8 = 0\), which simplifies to \(s - 8 = 0\).
03

Solve for the Variable

The equation \(s - 8 = 0\) needs to be solved for \(s\). Add 8 to both sides of the equation:\(s - 8 + 8 = 0 + 8\). Simplifying this, you get \(s = 8\).

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Key Concepts

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

Distributive Property
The distributive property is a crucial part of solving linear equations, especially when dealing with expressions inside parentheses. It allows you to expand expressions by distributing a factor across terms in parentheses.
In the given equation, which is \(2(s-4)=s\), the distributive property helps eliminate the parentheses first.
Here's how it works:
  • Multiply 2, which is outside the parentheses, by each term inside the parentheses separately.
  • This results in \(2 \times s - 2 \times 4 = 2s - 8\).
By doing this, you simplify the equation to \(2s - 8 = s\). Using the distributive property correctly simplifies equations and makes them easier to solve. This is an essential skill in Algebra that enables you to handle more complex equations with ease.
Isolate Variable
Isolating the variable, often labeled as \(s\), \(x\), or another letter of choice, is crucial for solving equations because it helps you find its value. In this process, we move all terms containing the desired variable to one side of the equation while maintaining balance.
Following the distribution in our example, the equation becomes \(2s - 8 = s\). Our goal now is to gather all variable terms (\(s\)) on one side:
  • Subtract \(s\) from both sides to get \(2s - s - 8 = 0\), which simplifies to \(s - 8 = 0\).
  • This move effectively isolates \(s\) on one side of the equation.
The equation now centers around \(s\), making it easier to solve. Isolating the variable is a strategy that simplifies the equation into a straightforward arithmetic problem.
Step-by-Step Solution
Breaking down a math problem into clear, logical steps is the essence of solving equations efficiently and avoiding mistakes. By focusing on incremental progress, we make challenging problems more digestible.Initially, we used the distributive property to open the equation and got \(2s - 8 = s\). Ensuring every action maintains the equation's balance is crucial:
  • Subtraction of \(s\) to concentrate it on one side simplifies it further to \(s - 8 = 0\).
  • From here, the solution becomes straightforward by adding 8 to both sides, solving \(s = 8\).
By following these steps, we systematically shrink a complex problem into smaller, actionable parts. Understanding each step and the reasoning behind it builds a solid foundation for tackling more formidable equations in the future.

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Most popular questions from this chapter

Set up an appropriate equation and solve. Data are accurate to two sig. digits unless greater accuracy is given. One lap at the Indianapolis Speedway is 2.50 mi. In a race, a car stalls and then starts \(30.0 \mathrm{s}\) after a second car. The first car travels at \(260 \mathrm{ft} / \mathrm{s},\) and the second car travels at \(240 \mathrm{ft} / \mathrm{s}\). How long does it take the first car to overtake the second, and which car will be ahead after eight laps?

Square an integer between 1 and 9 and subtract 1 from the result. Explain why the result is the product of the integer before and the integer after the one you chose.

Solve the given equations. All numbers are approximate. $$5.8-0.3(x-6.0)=0.5 x$$

Solve the given equations. All numbers are approximate. $$\frac{3.0}{7.0}=\frac{R}{42}$$

Perform the indicated divisions. The area of a certain rectangle can be represented by \(6 x^{2}+19 x+10 .\) If the length is \(2 x+5,\) what is the width? (Divide the area by the length.)
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