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Chapter 1: Problem 63

Solve. $$ 3(x+4)=7 x $$

Short Answer

Expert verified
x = 3

Step by step solution

01

- Distribute and Simplify

First, distribute the 3 on the left-hand side of the equation: 3(x+4) = 3x + 12. So the equation is now: 3x + 12 = 7x.
02

- Move Variables to One Side

Subtract 3x from both sides to move the variable terms to one side: 3x + 12 - 3x = 7x - 3x. Simplifying gives us: 12 = 4x.
03

- Isolate the Variable

Divide both sides by 4 to solve for x: \(\frac{12}{4} = \frac{4x}{4} \). Simplifying gives us: x = 3.

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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, used to simplify expressions and solve equations. In this property, you distribute, or 'spread,' the multiplication over each term inside the parentheses. For example, in the equation $$ 3(x+4)=7x $$, we need to apply the distributive property to the left side.This means multiplying each term inside the parentheses by 3: $$ 3 \times x + 3 \times 4 $$.So the equation becomes: $$ 3x + 12 = 7x $$.This step is crucial as it helps break down the equation into simpler parts, making it easier to solve.
Isolating Variables
Isolating the variable is the process of moving all instances of the variable to one side of the equation. This helps us in finding the value of the variable. After distributing the property, you had the equation: $$ 3x + 12 = 7x $$.To isolate the variable, we need to get rid of the variable term on one side. Subtract 3x from both sides to move all x terms to one side:$$ 3x + 12 - 3x = 7x - 3x $$.This simplifies to:$$ 12 = 4x $$.Now, we have all x terms on one side of the equation, making it easier to solve for x.
Simplifying Equations
Simplifying equations involves performing basic arithmetic operations until you have isolated the variable. Once all the terms with the variable are on one side, the next step is to isolate the variable completely. From the previous step, our simplified equation is: $$ 12 = 4x $$.To isolate x, divide both sides by 4:$$ \frac{12}{4} = \frac{4x}{4} $$.This leads to:$$ x = 3 $$.This shows the value of x that satisfies the original equation: $$ 3(x+4)=7x $$.Simplifying equations is the final step in solving linear equations, and it ensures that we've correctly isolated and solved for the variable.

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