Question

For what value of \(x\) is the equation \(2(x-6)+x=36\) true? A. 24 B. 16 C. 14 D. 10 E. 8

Short answer

Question: Solve the linear equation \(2(x-6) + x = 36\). Choose the correct answer from the given options: A) 12 B) 16 C) 20 D) 24 Answer: B) 16

Step-by-step solution

Simplify the equation

First, we need to distribute \(2\) across \((x-6)\). So, the equation becomes: \(2x - 12 + x = 36\).

Combine like terms in the equation

Combine the \(x\) terms and the constants on the left side of the equation. So, we have: \(2x + x - 12 = 36\), which simplifies to: \(3x - 12 = 36\).

Isolate the variable x

Add \(12\) to both sides of the equation. The equation then becomes: \(3x = 48\).

Solve for x

Divide both sides of the equation by \(3\). We get: \(x = 16\).

Match the answer with the given options

We found that the value of x is \(16\). Matching our answer with the given options, we see that the correct option is option B: \(16\).

Key concepts

Understanding Algebraic Expressions

When working with algebra, it's essential to become familiar with algebraic expressions. These are mathematical phrases that can include numbers, variables (like the letter x), and operations (such as addition and subtraction). In the exercise given, the expression 2(x-6) + x consists of variables and numbers combined with multiplication and addition.

It's important to grasp that the variable x represents an unknown value we're trying to find, and the number 2 is a coefficient, which multiplies the expression within the parentheses. Algebraic expressions can become quite complex, but the key to solving them often lies in simplifying and reshaping them into a more manageable form, as shown in the step-by-step solution.

Combining Like Terms

A crucial step in simplifying algebraic expressions is combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, in the equation 2x - 12 + x from our problem, 2x and x are like terms because they both contain the variable x to the power of one.

To combine them, simply add or subtract their coefficients: 2 and 1 (since x is the same as 1x) to get 3x. Remember that constants (like -12) can also be combined with other constants. This step is vital for finding the simplest form of an expression, making it easier to solve for the variable.

Isolating Variables

The end goal in solving linear equations is to find the value of the unknown variable, which in this case is x. To accomplish this, you'll need to isolate the variable on one side of the equation. This involves undoing the operations that are being performed on the variable.

In our exercise, after combining like terms, we do this by first adding 12 to both sides to cancel out the -12 and then divide everything by 3, the coefficient of x. The variable x is considered isolated when it's by itself on one side of the equation, with a coefficient of 1. At this point, you'll have a clear answer: in our case, x equals 16.