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Chapter 11: Problem 47

Solve. $$ 40=5(3 x+2) $$

Short Answer

Expert verified
x = 2

Step by step solution

01

Distribute the 5

Expand the right-hand side of the equation by distributing the 5 into the parentheses.\[ 40 = 5 \times 3x + 5 \times 2 \]This gives:\[ 40 = 15x + 10 \]
02

Subtract 10 from both sides

Eliminate the constant term on the right-hand side by subtracting 10 from both sides of the equation.\[ 40 - 10 = 15x + 10 - 10 \]This simplifies to:\[ 30 = 15x \]
03

Divide both sides by 15

Solve for \( x \) by dividing both sides of the equation by 15.\[ \frac{30}{15} = \frac{15x}{15} \]This simplifies to:\[ x = 2 \]

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Key Concepts

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

Distributive Property
Understanding the distributive property is crucial for solving linear equations. In essence, it allows you to multiply a term outside the parentheses by each term inside the parentheses. Consider the exercise we have:

$$40 = 5(3x + 2)$$

To distribute the 5, multiply it by both 3x and 2. This means:
  • $$5 \times 3x = 15x$$
  • $$5 \times 2 = 10$$
Thus, our equation becomes:
$$40 = 15x + 10$$

This step helps to simplify the equation by removing the parentheses, allowing us to work with individual terms.
Eliminating Constants
The next step is to isolate the variable term by eliminating the constants. In our simplified equation:

$$40 = 15x + 10$$

The constant is 10 on the right-hand side. We need to move this to the left-hand side to start isolating the variable term. Do this by subtracting 10 from both sides:

$$40 - 10 = 15x + 10 - 10$$

This simplifies to:
$$30 = 15x$$

Now, the equation is much simpler and closer to solving for x.
Division in Equations
Finally, to solve for the variable x, we need to use division. In the equation:

$$30 = 15x$$

We want to isolate x by getting it alone on one side of the equation. Divide both sides by 15:

$$\frac{30}{15} = \frac{15x}{15}$$

This yields:

$$2 = x$$

Or more commonly written as:
$$x = 2$$

The concept of division in equations helps to simplify and isolate the variable, providing the final solution.

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