Problem 76 - Elementary Algebra

Question

Find the solution. If twenty-one is subtracted from some number and that result is multiplied by two, the result is thirty-eight. What is the number?

Step-by-Step Solution

Answer

Answer: The correct number is 40.

Step 1: Write the equation

We are given that if 21 is subtracted from a number and the result is multiplied by 2, the result is 38. Let the number be x. We can write the equation as: 2(x-21) = 38.

Step 2: Use the distributive property

To solve for x, we need to simplify the equation. First, we will apply the distributive property to get rid of the parentheses: 2 * (x-21) = 2x - 42. Now, our equation looks like this: 2x - 42 = 38.

Step 3: Add 42 to both sides

Next, we will isolate the term with the variable (2x) by adding 42 to both sides of the equation: (2x - 42) + 42 = 38 + 42. This simplifies to: 2x = 80.

Step 4: Divide by 2

Finally, we need to solve for x by dividing both sides of the equation by 2: (2x)/2 = 80/2. This simplifies to: x = 40.

Step 5: Check the solution

To check if our solution is correct, we can substitute x back into the original equation and see if both sides are equal: 2(40-21) = 38. This simplifies to 2(19) = 38, which is true. Therefore, the correct number is x = 40.

Key Concepts

Equations

In algebra, an equation is a mathematical statement that shows the equality of two expressions. It typically involves variables, numbers, and operations. An equation is like a balance, where you need both sides to weigh the same. In our exercise, we are given a word problem that hints at an equation we can write. The information tells us: "If twenty-one is subtracted from some number and that result is multiplied by two, the result is thirty-eight." Here, we can translate this description into an algebraic equation. An "unknown" number is often represented with a variable like \( x \). The statement translates into the equation: \( 2(x - 21) = 38 \). This equation indicates a specific relationship among the numbers involved and \( x \). It shows us what we need to solve to find our unknown number.

Distributive Property

The distributive property is a useful algebraic rule for simplifying expressions, especially when dealing with parentheses. It allows you to multiply a single term by two or more terms inside the parentheses. In the context of our equation \( 2(x - 21) = 38 \), applying the distributive property means we multiply the number outside the parentheses, in this case 2, by each term inside the parentheses. When we distribute:

• Multiply 2 by \( x \)
• Multiply 2 by \(-21\)

This results in the expression \( 2x - 42 \). By using the distributive property, we can simplify the expression and make the equation easier to work with. This step is crucial because it prepares the equation for the next phase, solving for the variable, \( x \).

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. After using the distributive property, our equation becomes \( 2x - 42 = 38 \). To solve this, we perform operations to get \( x \) alone on one side of the equation. Here’s the step-by-step process:

• Add 42 to both sides to eliminate the -42: \( 2x - 42 + 42 = 38 + 42 \), simplifying to \( 2x = 80 \).
• Divide both sides by 2 to isolate \( x \): \( \frac{2x}{2} = \frac{80}{2} \), simplifying to \( x = 40 \).

These operations help "undo" the operations applied to \( x \), gradually isolating it. Solving an equation often involves reversing arithmetic operations using opposite operations, like addition for subtraction or division for multiplication. Finally, once you find a solution, it’s always a good idea to substitute it back into the original equation to verify its correctness. In this exercise, substituting \( x = 40 \) confirms both sides of the equation equal 38, proving our solution is correct.