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Chapter 3: Problem 20

Solve each equation. $$6(t+5)=2(3 t+15)$$

Short Answer

Expert verified
The solution is all real numbers.

Step by step solution

01

Expand Both Sides

Start by expanding both sides of the equation to eliminate parentheses. The equation is:\[ 6(t+5) = 2(3t + 15) \] Expand both sides by distributing the numbers outside the parentheses:\[ 6t + 30 = 6t + 30 \]
02

Simplify Equation

Look at the resulting equation from Step 1:\[ 6t + 30 = 6t + 30 \]Both sides are identical, which means they cancel each other out.
03

Analysis of the Result

Since both sides of the equation are the same, this shows that the equation is true for all values of \(t\). There are no specific constraints, so the solution is all real numbers.

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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 concept in algebra that helps simplify equations and expressions. It involves multiplying a single term by each term inside a set of parentheses.

The rule is often expressed as:
  • \(a(b + c) = ab + ac\)
This property enables us to remove parentheses and simplify expressions by distributing or spreading the multiplication across each term in the sum or difference within the parentheses.

In our example equation, \(6(t + 5) = 2(3t + 15)\), applying the distributive property, we have:
  • Expand the left side: \(6t + 30\)
  • Expand the right side: \(6t + 30\)
This operation simplifies both sides of the equation and shows us that they are identical, which is key in solving the equation.
Identity Equation
An identity equation is an equation that is always true, no matter the value of the variable involved.

It occurs when both sides of the equation are identical after simplification. This means that every real number is a solution to the equation.

For example, in the equation \(6t + 30 = 6t + 30\), both sides are exactly the same. This showcases an identity equation.

When solving algebraic equations, if you find that the simplified version results in an identity, it implies:
  • Every substitution of the variable makes the equation true.
  • There is not just one solution, but infinite solutions.
Understanding identity equations helps recognize when problems involve such infinite solutions, rather than singular solutions.
Real Numbers
Real numbers encompass a vast set of numbers used commonly in mathematics. They include:
  • All positive and negative numbers
  • Fractions
  • Decimals
  • Zero
Real numbers are typically represented by the letter \(\mathbb{R}\) and make up the vast majority of numbers we encounter in everyday math problems.

In the context of our exercise, since the equation \(6t + 30 = 6t + 30\) holds true for any value of \(t\), all real numbers serve as solutions to this identity equation.

Recognizing that real numbers are the solution set for identity equations highlights the completeness of this numerical system. It showcases the utility of real numbers in providing solutions to algebraic equations without limitations.

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Most popular questions from this chapter

Consider these two solutions: $$ \begin{array}{rrr} 3(x+2)=9 & 3(x-4)=7 \\ \frac{3(x+2)}{3}=\frac{9}{3} & 3 x-12=7 \\ x+2=3 & 3 x=19 \\ x=1 & x=\frac{19}{3} \end{array} $$ Are both of these solutions correct? Comment on the effectiveness of the two different approaches.

Solve each equation. $$\frac{1}{2} x-\frac{4}{3}=-\frac{5}{6}$$

Solve each equation. $$\frac{x-1}{4}+\frac{x+2}{5}=\frac{39}{20}$$

Mario has a collection of 22 specimens in his aquarium consisting of crabs, fish, and plants. There are three times as many fish as crabs. There are two more plants than crabs. How many specimens of each kind are in the collection?

Maida has 18 coins consisting of dimes and quarters amounting to \(\$ 3.30\). How many coins of each kind does she have?
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