Problem 50 - Algebra A Combined Function

Question

Solve each equation. $$ 3 x-7=11 $$

Step-by-Step Solution

Answer

The solution is x = 6.

Step 1: Identify the Equation

The given equation to solve is: \[ 3x - 7 = 11 \] Our objective is to find the value of \( x \) that satisfies this equation.

Step 2: Isolate the Variable Term

To isolate the term with \( x \), we need to eliminate the constant on the left side of the equation. Add 7 to both sides of the equation to do this:\[ 3x - 7 + 7 = 11 + 7 \]Simplifying both sides, we have:\[ 3x = 18 \]

Step 3: Solve for the Variable

Now that the equation is simplified to \( 3x = 18 \), we need to solve for \( x \). Divide both sides of the equation by 3 to isolate \( x \):\[ \frac{3x}{3} = \frac{18}{3} \]Simplifying, we find:\[ x = 6 \]

Key Concepts

Solving Equations

Solving equations is like being a detective, and our main aim is to uncover the unknown value that makes the equation true. In algebra, equations are statements that assert the equality of two expressions. Our goal is to find the value of the unknown, typically represented by a variable like \( x \), that makes both sides of the equation equal.
To solve an equation, we need to transform it step by step into a simpler, more manageable form. This usually means performing operations on both sides of the equation until the variable is isolated, revealing its value.
It's important to apply the same operation to both sides of the equation to maintain the balance or equality.

Isolation of Variables

Isolating the variable is often the first milestone on the journey to solving an equation. This concept is all about rearranging the equation to get the variable on one side by itself.
Consider our original equation: \( 3x - 7 = 11 \). Here, \( 3x \) is the term with the variable, and \(-7\) is a constant. We need to move constants away from the variable. In this equation, we do that by adding 7 to both sides:

• \( 3x - 7 + 7 = 11 + 7 \)

Simplifying gives:

• \( 3x = 18 \)

With the constant removed from the left side, we have successfully isolated the variable term \( 3x \). The next step will reveal the value of \( x \).

Simplification Steps

Once we've isolated the variable term, our task is to simplify the equation to solve for the variable. Simplification involves reducing the equation step by step until the solution is found.
After isolating \( 3x \) in the equation \( 3x = 18 \), we divide each side by 3 to get \( x \) by itself:

• \( \frac{3x}{3} = \frac{18}{3} \)

This division simplifies as:

• \( x = 6 \)

At this stage, we've solved the equation. The key to these steps is performing basic arithmetic operations that gradually reveal the solution, ensuring we do the same operation on both sides to keep the equation balanced.
Factors and divisors are crucial here, as they help to break down the numbers, making it straightforward to find the answer. Simplifying equations might seem tricky at first, but with practice, it becomes an intuitive process.