Problem 38 - Algebra 2

Question

For Exercises \(36-38,\) use the following information. Damaso asked Emilia to choose a number between 1 and \(35 .\) He told her to subtract 12 from that number, multiply by \(2,\) add \(10,\) and divide by \(4 .\) Emilia's final number was \(9 .\) What was her original number?

Step-by-Step Solution

Answer

The original number was 25.

Step 1: Set Up the Equation

Translate Emilia's process into an equation. Let the original number be \( x \). According to the problem: 1. Subtract 12: \( x - 12 \)2. Multiply by 2: \( 2(x - 12) \)3. Add 10: \( 2(x - 12) + 10 \)4. Divide by 4: \( \frac{2(x-12) + 10}{4} \)The result is given as 9.

Step 2: Simplify the Equation

Use algebra to set up the equation from Step 1 as follows:\[ \frac{2(x-12) + 10}{4} = 9 \]

Step 3: Eliminate the Denominator

Multiply both sides of the equation by 4 to eliminate the fraction:\[ 2(x-12) + 10 = 36 \]

Step 4: Simplify Further

Distribute the 2 on the left-hand side:\[ 2x - 24 + 10 = 36 \] Simplify:\[ 2x - 14 = 36 \]

Step 5: Solve for x

Add 14 to both sides to isolate the term with \(x\):\[ 2x = 50 \] Then, divide both sides by 2 to solve for \(x\):\[ x = 25 \]

Step 6: Verify the Solution

Substitute \( x = 25 \) back into Emilia's process:1. Subtract 12: \( 25 - 12 = 13 \)2. Multiply by 2: \( 13 \times 2 = 26 \)3. Add 10: \( 26 + 10 = 36 \)4. Divide by 4: \( \frac{36}{4} = 9 \)The final number matches the given result, confirming our solution is correct.

Key Concepts

Problem Solving

Problem solving in algebra involves translating a real-world scenario into mathematical language. When Damaso asked Emilia to perform a series of operations on a chosen number, our task was to determine the original number using those operations.
To solve problems like this, we often take the following steps:

• Identify the operations given in a problem.
• Translate these operations into a mathematical equation.
• Work through the problem step by step using algebraic techniques.

In this particular exercise, we translated Emilia's operations into a sequence of actions: subtracting, multiplying, adding, and dividing.
Each step built on the previous one until we reached the final given result. By setting this up as an algebraic equation, we can systematically work backward to find the initial number.

Linear Equations

Linear equations are a foundational concept in algebra. They involve variables represented in a way that forms a straight line when plotted on a graph. In our exercise, we formed a linear equation to find Emilia's original number.
The equation from the problem \[ \frac{2(x-12) + 10}{4} = 9 \]was initially a bit complex due to its fractional form. Our task was to simplify it step by step so that we could solve for the unknown variable.
Key characteristics of linear equations include:

• Constant rate of change.
• The power of the variable is always one.
• Graph is a straight line.

Solving linear equations often involves isolating the variable by performing operations that maintain its equality, such as addition, subtraction, multiplication, and division.

Equation Simplification

Equation simplification is about reducing an equation to its most straightforward form to make solving it easier. In this exercise, we started with a more complex equation and simplified it using algebraic operations.
Here is how we approached simplification:

• Eliminate fractions by multiplying both sides by the denominator.
• Use the distributive property to remove parentheses.
• Combine like terms to make the equation more manageable.
• Isolate the variable to the left of the equality sign.

For example, eliminating the denominator in \[ \frac{2(x-12) + 10}{4} = 9 \]required multiplying both sides by 4. This step transformed the equation into \\[ 2(x-12) + 10 = 36 \]which is simpler and more straightforward.
The goal of simplification is to reveal the simplest version of an equation that still holds the same relationships, making it easier to solve and verify the solution.