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Chapter 2: Problem 79

\(5(3 x-2)-14 x=3\)

Short Answer

Expert verified
x = 13.

Step by step solution

01

Distribute the 5

Expand the expression by distributing the 5 in the equation: 5(3x - 2) - 14x = 3. This gives: 15x - 10 - 14x = 3.
02

Combine like terms

Combine the terms with the variable x: 15x - 14x - 10 = 3. This simplifies to: x - 10 = 3.
03

Solve for x

Add 10 to both sides of the equation to isolate x: x - 10 + 10 = 3 + 10. This gives: x = 13.

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Key Concepts

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

Distributing in Algebra
When solving algebraic equations, you often encounter expressions with parentheses. These need to be simplified before solving the equation. This process is called 'distributing'.
Distributing involves multiplying each term inside the parentheses by the factor outside.
In our example, we start with the equation:

5(3x - 2) - 14x = 3.

To distribute, multiply 5 by each term inside the parentheses:

5 * 3x = 15x
5 * -2 = -10

This transforms the equation into:

15x - 10 - 14x = 3.

Distributing is crucial as it helps break down the equation into simpler parts, making it easier to handle.
Combining Like Terms
After distributing, the next step is to simplify the equation by combining like terms. 'Like terms' are terms that contain the same variable raised to the same power.

From our distributed expression:

15x - 10 - 14x = 3,

we combine the terms with the variable x. That is 15x and -14x:

15x - 14x = x

Now we have:

x - 10 = 3.

Combining like terms reduces the equation to a simpler form, which is crucial for the next step鈥攊solating the variable.
Isolating Variables
Isolating the variable is the final step in solving a linear equation. The goal is to get the variable (in this case, x) on one side of the equation to find its value.

From the previous step, we have:

x - 10 = 3.

To isolate x, add 10 to both sides of the equation:

x - 10 + 10 = 3 + 10.

This gives:

x = 13.

By following the steps of distributing, combining like terms, and isolating the variable, we successfully solved the equation and found that x = 13.

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