Problem 151 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation. Round your answer to three decimal places, if necessary. $$3 x-7=14$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 7\).

Step 1: Isolate the Variable

Start by adding 7 to both sides of the equation \(3x - 7 = 14\). It will give \(3x = 21\).

Step 2: Solve for x

Divide both sides of the equation by 3 to solve for \(x\). It will result in \(x = 7\).

Step 3: Verify the Solution

Substitute \(x = 7\) back into the original equation \(3x - 7 = 14\) to verify the solution. It gives \(21 - 7 = 14\) which is correct.

Key Concepts

Solving Linear Equations

Solving linear equations is like solving a puzzle. The goal is to find the value of the variable that makes the equation true. A linear equation is an equation in which each term is either a constant or the product of a constant and a single variable. Equations like \(3x - 7 = 14\) are typical linear equations. Here, our job is to determine the value of \(x\) in a straightforward and methodical way.
\[\]Let's break it down into clear steps:

• Look at both sides of the equation. Identify the operations applied to the variable you need to solve for.
• Use inverse operations to find the value of the variable. This means doing the opposite operation to cancel out terms and simplify the equation.

By following these steps, you carefully balance the equation and gradually move the variable to one side while all other numbers go to the other side.

Isolation of Variables

The core goal in simplifying any equation is to isolate the variable. This means getting the variable by itself on one side of the equation. Only then can you clearly see its value or relationship to other numbers. In our sample equation \(3x - 7 = 14\), "isolating" \(x\) involves a couple of key moves:
\[\]

• Addition or Subtraction: Start by adding or subtracting terms to both sides of the equation that don’t involve the variable. This helps move constants to the opposite side.
• Multiplication or Division: Once all additive or subtractive constants are moved, you can multiply or divide to finalize isolation. In our example, after getting \(3x = 21\), divide by 3 to give \(x = 7\).

This method is like peeling an onion, layer by layer until you reach the core variable.

Verification of Solutions

Verifying the solution is the final essential step in solving equations. Even if you think you've found the correct answer, it's crucial to confirm it is correct by substituting the solution back into the original equation.
\[\]Here's how verification works:

• Take the value you found for the variable and plug it back into the original equation.
• Calculate each side separately to ensure both sides of the equation are equal.

For instance, with \(x = 7\) in the original equation \(3x - 7 = 14\), substitute 7 for \(x\). You get \(3(7) - 7 = 14\). This equals \(21 - 7 = 14\), confirming that the solution is valid. Verification not only strengthens accuracy but also boosts your confidence in your algebraic skills.