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Chapter 1: Problem 56

\(2 r+6=8 ; \quad 1\)

Short Answer

Expert verified
r = 1

Step by step solution

01

Isolate the variable term

First, subtract 6 from both sides of the equation to isolate the term with the variable. This gives us:\[ 2r + 6 - 6 = 8 - 6 \]Which simplifies to:\[ 2r = 2 \]
02

Solve for the variable

Next, divide both sides of the equation by 2 to solve for the variable \(r\). This gives us:\[ \frac{2r}{2} = \frac{2}{2} \]Which simplifies to:\[ r = 1 \]

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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 is often to isolate the variable. This means getting the variable (like 'r' in our example) on one side of the equation with no other terms.

To do this, perform operations that will move constants or coefficients to the other side of the equation. In the exercise, the equation starts as:

2r + 6 = 8
To isolate 'r', you would subtract 6 from both sides:

2r + 6 - 6 = 8 - 6

This simplification leaves us with:

2r = 2

The term with the variable 'r' is now isolated on one side.
Simplification
Once you have isolated the variable term, the next step is to simplify the equation as much as possible. Simplification means reducing the equation to its simplest form.

In our equation, after subtracting 6 from both sides, we have:

2r = 2

This is already quite simple, but if it wasn't, you'd combine like terms and perform any necessary arithmetic to further reduce the equation. Always remember, simplification helps in making the final steps much easier.
Division in Algebra
In algebra, division is a crucial operation used to solve for variables. When a variable is multiplied by a coefficient, you can isolate the variable by dividing both sides of the equation by that coefficient.

In our exercise, we reached the step:

2r = 2

To solve for 'r', divide both sides by the coefficient of 'r', which is 2:

\( \frac{2r}{2} = \frac{2}{2} \)

This division gives us a much simpler result:

\( r = 1 \)

Remember, division allows you to simplify equations and isolate the variable, which makes finding the solution straightforward.

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Most popular questions from this chapter

Solve each problem. In \(2006,\) bachelor's degree recipients from private four-year institutions had an average of \(\$ 18,800\) in total student debt. This average increased \(\$ 1400\) by 2011 and then dropped \(\$ 300\) by \(2016 .\) What was the average amount of total student debt in \(2016 ?\) (Data from The College Board.)

Use the distributive property to rewrite each expression. $$ -\frac{1}{4}(8 x+3) $$

Write a numerical expression for each phrase, and simplify the expression. The sum of -3 and 5 and -12

A manufacturer has fixed costs of \(\$ 1000\) to produce gizmos. Each gizmo costs \(\$ 5\) to make. The fixed cost to produce gadgets is \(\$ 750,\) and each gadget costs \(\$ 3\) to make. Write an expression for the cost to make \(y\) gadgets.

Use the distributive property to rewrite each expression. $$ -3(2 x-5) $$
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