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Chapter 2: Problem 63

Solve each equation, if possible. $$ -(19-3 s)-(8 s+1)=35 $$

Short Answer

Expert verified
The solution to the equation is \(s = -11\).

Step by step solution

01

Distribute the negative signs

First, distribute the negative signs in the equation \[-(19 - 3s) - (8s + 1) = 35. \] This becomes:\[-19 + 3s - 8s - 1 = 35.\] Combine like terms to simplify the equation.
02

Combine like terms

Now, combine the terms with \(s\) and the constant terms. So, we have:\[(3s - 8s) + (-19 - 1) = 35.\] Simplify it as:\[-5s - 20 = 35.\]
03

Isolate the term with the variable

Add 20 to both sides to isolate the term with the variable on the left side:\[-5s - 20 + 20 = 35 + 20,\]which simplifies to:\[-5s = 55.\]
04

Solve for the variable

Divide both sides of the equation by -5 to solve for \(s\):\[s = \frac{55}{-5}.\] Simplify the fraction:\[s = -11.\]

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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 fundamental tool in algebra. It helps you to simplify expressions and solve equations by distributing or spreading a single term across terms inside parentheses. This is especially useful when dealing with linear equations containing parentheses.

When faced with the expression \(-(19 - 3s)\) for example, you need to apply the distributive property to eliminate the parentheses. Here's how:
  • Multiply the terms inside the parentheses by the negative sign outside.
  • This flips the sign of each term, resulting in \(-19 + 3s\).
The same principle applies to \(-(8s + 1)\), turning it into \(-8s - 1\).

By correctly using the distributive property, you can transform a complex equation into a much simpler form that鈥檚 easier to solve.
Combining Like Terms
Once we've distributed terms correctly, it's important to simplify the equation by combining like terms. Like terms in algebra are terms that contain the same variable raised to the same power.

In our example, after applying the distributive property, the equation looks like:
  • \(-19 + 3s - 8s - 1 = 35\).
To combine like terms:
  • Group the variable terms together \((3s - 8s)\).
  • Group the constant terms together \((-19 - 1)\).
  • Simplify these groups: \(-5s - 20 = 35\).
This step reduces the complexity of the equation, making it more straightforward to isolate the variable and find its value.
Solving Linear Equations
To solve linear equations, the goal is to isolate the variable on one side of the equation. This allows us to find the solution for that variable.

Continuing with our formula:
  • Start from \(-5s - 20 = 35\).
  • Add 20 to both sides to move the constants to one side: \(-5s = 35 + 20\).
  • This simplifies the equation to \(-5s = 55\).
Now, address the variable's coefficient:
  • To isolate \(s\), divide every term by \(-5\).
  • So, \(s = \frac{55}{-5}\) simplifies to \(s = -11\).
Understanding each step ensures that you solve linear equations correctly and efficiently, leading to accurate results without confusion.

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