Problem 14 - Elementary Algebra

Question

Solve. $$2 x x+5=16-x$$

Step-by-Step Solution

Answer

\( x = \frac{11}{3} \).

Step 1: Identify Like Terms

First, let's determine all the terms with the variable \( x \) and constants. The equation is \( 2x + 5 = 16 - x \). We can combine like terms on either side of the equation.

Step 2: Move Terms with \( x \) to One Side

Add \( x \) to both sides to move all terms involving \( x \) to the left side. This gives us:\[ 2x + x + 5 = 16 \].

Step 3: Simplify Left Side

Combine the \( x \) terms on the left side. \( 2x + x \) becomes \( 3x \). The equation now looks like:\[ 3x + 5 = 16 \].

Step 4: Isolate Terms with \( x \)

Subtract 5 from both sides to isolate the term with \( x \):\[ 3x + 5 - 5 = 16 - 5 \],which simplifies to:\[ 3x = 11 \].

Step 5: Solve for \( x \)

Divide both sides by 3 to solve for \( x \):\[ x = \frac{11}{3} \].

Key Concepts

Like Terms

In algebra, "like terms" are terms that have the same variable raised to the same power. In the equation we're solving, you notice that each side has terms with the variable \( x \) and constants. Recognizing and combining like terms is crucial because it helps simplify equations, making them easier to solve.
When looking at the equation \( 2x + 5 = 16 - x \), the terms \( 2x \) and \( -x \) are like terms because they both contain the variable \( x \). This is why, in the solution steps, we focus on combining these terms to simplify our equation.

• Only terms with the same variable and exponent can be combined.
• By combining like terms, you reduce the number of terms, simplifying the equation.

Isolate Variable

To solve linear equations, isolating the variable is an essential step. This means you rearrange the equation to get the variable alone on one side of the equation. In our equation, we want to isolate \( x \).
To accomplish this, we moved all terms involving \( x \) to one side. Adding \( x \) to both sides was a strategic step to collect all \( x \) terms on one side of the equation. Post-simplification, the equation \( 3x + 5 = 16 \) contains only one term with \( x \), so it's easier to isolate it further.
To completely isolate \( x \), we subtract constants from both sides, resulting in \( 3x = 11 \). This leaves \( x \) ready to be isolated by simple division.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In the problem \( 2x + 5 = 16 - x \), the left and right sides are algebraic expressions. Understanding how to manipulate algebraic expressions is key for solving equations.

• An algebraic expression can be as simple as a single variable or more complex with multiple terms and operations.
• By manipulating expressions, such as combining like terms and simplifying, you create a clearer path to solve for variables.

In the given equation, each part of \( 2x + 5 \) and \( 16 - x \) represents an algebraic expression. Getting familiar with the structure of algebraic expressions helps in simplifying and solving equations.

Solving Equations

Solving equations is the process of finding the value(s) of the variable(s) that make the equation true. In the example \( 2x + 5 = 16 - x \), our goal was to determine the value of \( x \).
Here’s a step-by-step approach to solving equations like this:

• Identify and combine like terms.
• Move all terms containing the variable to one side of the equation.
• Simplify both sides as needed.
• Isolate the variable by performing operations like addition, subtraction, multiplication, or division.
• Check your solution by substituting the value back into the original equation.

In our example, after following these steps, we ended up with \( x = \frac{11}{3} \), which is then checked to ensure it satisfies the original equation.