Problem 39 $3(x+4)=3 x+3(4)$... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 1: Problem 39

$3(x+4)=3 x+3(4)$

Short Answer

Expert verified

The equation is an identity.

Step by step solution

- Distribute 3 on the Left Side

Distribute the 3 to both terms inside the parentheses on the left side of the equation: 3(x + 4) = 3x + 12

- Simplify Both Sides

Notice that the right side of the equation is already simplified: 3x + 12 = 3x + 12

- Compare Both Sides

Both sides of the equation are equal: 3x + 12 = 3x + 12. This means the equation is an identity and holds true for all values of x.

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Key Concepts

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

Solving Linear Equations

Solving linear equations is one of the fundamental skills in algebra. A linear equation is an equation that makes a straight line when graphed. It typically has the form $ax + b = c$, where $a$, $b$, and $c$ are constants. To solve, you need to find the value of $x$ that makes the equation true.
Here鈥檚 how you approach it:

Step 1: Combine like terms on both sides.
Step 2: Move all the terms involving $x$ to one side.
Step 3: Move constant terms to the opposite side.
Step 4: Divide or multiply as needed to isolate $x$.

In the given exercise, we first used the distributive property to expand the expression, then simplified it. Since both sides of the equation were the same, we concluded it was an identity, meaning it's true for all values of $x$.

Simplifying Expressions

Simplifying expressions involves combining like terms and using arithmetic to condense the expression into its simplest form. This is important as it makes solving equations easier and more manageable.
For example, in the exercise:

$3(x + 4) = 3x + 3 \cdot 4$: The distributive property is used to multiply 3 by both $x$ and 4, which gives $3x + 12$.
The right-hand side was already $3x + 12$, so no further simplification was needed there.

This simplified forms tells us that the expression behaves consistently across both sides, which leads directly into understanding algebraic identities.

Identities in Algebra

An identity in algebra is an equation that is always true, no matter what values are substituted for the variables. This means that both sides of the equation are equivalent.
For instance, in the exercise we had:

Left side after distribution: $3(x + 4) = 3x + 12$
Original right side: $3x + 12$

Since both sides simplified to the same expression, $3x + 12 = 3x + 12$, we recognized it as an identity.
Identities help in confirming that your manipulations and simplifications are correct. They're also useful in solving more complex problems where confirming consistency can save time and effort.

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91影视

\(3(x+4)=3 x+3(4)\)

Short Answer

Step by step solution

- Distribute 3 on the Left Side

- Simplify Both Sides

- Compare Both Sides

Key Concepts

Solving Linear Equations

Simplifying Expressions

Identities in Algebra

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