Problem 18 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 7 and 8 . \(3 x-1=26\)

Step-by-Step Solution

Answer

The solution is \(x = 9\).

Step 1: Identify the Equation

We start with the equation given: \(3x - 1 = 26\). This is a simple linear equation in one variable \(x\). Our task is to find the value of \(x\) that satisfies this equation.

Step 2: Isolate the Variable Term

To isolate the term with the variable \(x\), we first need to eliminate the constant term on the left side. We do this by adding 1 to both sides of the equation:\[3x - 1 + 1 = 26 + 1\]Simplifying both sides gives us:\[3x = 27\]

Step 3: Solve for the Variable

Now that we have \(3x = 27\), we need to solve for \(x\) by dividing both sides of the equation by 3:\[\frac{3x}{3} = \frac{27}{3}\]Simplifying further results in:\[x = 9\]

Step 4: Check the Solution

To verify our solution, we substitute \(x = 9\) back into the original equation:\[3(9) - 1 = 26\]Calculate the left side:\[27 - 1 = 26\]Since both sides of the equation are equal, our solution \(x = 9\) is correct.

Key Concepts

Solving Equations

When faced with solving equations, especially linear ones like the equation \(3x - 1 = 26\), our goal is to find the value of the unknown variable that makes the equation true. Each step aims to make the equation simpler. Breaking down the process:

• Start by identifying the equation and recognizing its type; here, it's a linear equation involving a single variable, \(x\).
• Look to isolate the variable by moving other terms to the opposite side. In this example, adding 1 to both sides transforms \(3x - 1 = 26\) into \(3x = 27\).
• Finally, divide each side by the coefficient of \(x\). Since \(x\) is multiplied by 3, you divide by 3 to find \(x = 9\).

By simplifying step-by-step, you solve the equation efficiently. The process is a puzzle where each step brings you closer to the solution.

Checking Solutions

Checking your solutions ensures you haven't made any errors in your calculations and that the solution is valid. It's a critical step:

• Substitute your found value back into the original equation. For \(x = 9\) in \(3x - 1 = 26\), replace \(x\) with 9.
• Calculate both sides. If they match, as in \(3(9) - 1 = 26\) transforms into \(27 - 1 = 26\), your solution is verified.
• This step confirms that the solution meets all requirements of the original equation.

A correct solution balances the equation, showing both logic and arithmetic were followed precisely. It's like double-checking your homework to ensure you didn't overlook anything.

Single Variable Equations

Single variable equations involve only one unknown value, making them straightforward to solve. They're a common part of foundational algebra:

• Such equations typically take the form \(ax + b = c\), where the goal is to find \(x\).
• The solution involves performing operations to isolate \(x\). For instance, moving all terms except those attached to \(x\) across the equation is key.
• After isolating, solve simple arithmetic to determine \(x\). This process helps build systematic problem-solving skills.

Understanding single variable equations helps you in more complex algebraic problems where similar principles apply. They're often the building blocks for tackling real-world math scenarios.