Problem 66 - Elementary Algebra

Question

Solve. $$ 34-16(4 x-9)=2 $$

Step-by-Step Solution

Answer

x = 2.75

Step 1: Distribute the 16

First, distribute the 16 into the parentheses. This means you multiply 16 by each term inside the parentheses: \[ 34 - (16 imes 4x) + (16 imes 9) = 2 \] This simplifies to: \[ 34 - 64x + 144 = 2 \] since \(16 imes 4x = 64x\) and \(16 imes 9 = 144\).

Step 2: Combine like terms

Next, combine the constant terms on the left side of the equation. \[ (34 + 144) - 64x = 2 \] This simplifies to: \[ 178 - 64x = 2 \].

Step 3: Move constant to the right side

Isolate the term with \(x\) by subtracting 178 from both sides of the equation: \[ -64x = 2 - 178 \] Simplifying gives: \[ -64x = -176 \].

Step 4: Solve for x

Finally, solve for \(x\) by dividing both sides by -64: \[ x = \frac{-176}{-64} \] This simplifies to: \[ x = \frac{176}{64} \] which further simplifies to \[ x = \frac{11}{4} \] or \( x = 2.75 \).

Key Concepts

Distributive Property

The distributive property is a fundamental aspect of algebra that allows you to break down expressions into simpler parts. It is particularly useful when you encounter expressions involving parentheses. In our original exercise, we have to distribute the number 16 to both terms inside the parentheses:

• First, multiply 16 by the term containing the variable, which is 4x. This results in 16 times 4x, equaling 64x.
• Next, multiply 16 by the constant term 9, which results in 16 times 9, equaling 144.

So, for the original expression 34 - 16(4x - 9), distributing the 16 gives us 34 - 64x + 144.
This property is crucial for simplifying and rewriting expressions, and this step ensures that each term inside the parentheses is affected by the number outside.

Combining Like Terms

Combining like terms is a method used to simplify expressions or equations. Like terms are terms that contain the same variables raised to the same power. Once the distributive property is applied, combining like terms helps organize the expression further.
In our example, after applying the distributive property, the equation becomes 34 - 64x + 144. The next step is to combine any constant terms on the left side:

• The constants here are 34 and 144.
• Add these numbers together to combine them into a single constant term.

So, when you add 34 and 144, you get 178.
This simplifies the equation to 178 - 64x = 2, making it clearer and easier to solve for the variable.
Remember, combining like terms allows you to condense the equation and focus on the variable terms.

Solving Linear Equations

Once the equation is simplified, the next step is to solve it, which involves isolating the variable. Solving linear equations requires a series of operations to find the value of the unknown variable, often denoted by x.
In the given problem, after simplifying, we have 178 - 64x = 2. Here's what you need to do:

• Move the constant term to the other side by subtracting 178 from both sides: -64x = 2 - 178.
• Simplify the right-hand side to get -64x = -176.
• Divide both sides by the coefficient of x, which is -64, to solve for x: x = \( \frac{-176}{-64} \).

This calculation shows x = \( \frac{11}{4} \) or 2.75.
This process emphasizes clear, step-by-step manipulation to uncover the value of x.
It's crucial to perform inverse operations systematically to keep the equality balanced.