Problem 35 - Algebra 1: Concepts and Skills

Question

Write the sentence as an equation. Let x represent the number. Use mental math to solve the equation. Then check your solution. The quotient of 49 and a number is 7.

Step-by-Step Solution

Answer

The number represented by x is 7.

Step 1: Translate the word problem into an algebraic equation

The key in this step is to translate the words meaningfully into algebraic form. From the word problem 'The quotient of 49 and a number is 7', we derive the equation \( \frac{49}{x} = 7 \).

Step 2: Solve the equation

Here, we need to isolate x to solve for it. First, multiply both sides of the equation by x in order to cancel it out on the left side of the equation. The equation becomes: \( 49 = 7x \). Then, divide each side of the equation by 7 to isolate x. The final algebraic solution is: \( x = 7 \).

Step 3: Check your solution

It's important to substitute the solution into the original equation to verify its accuracy. By plugging x = 7 into \(\frac{49}{x} = 7\), it is observed that both sides of the equation equal 7, hence confirming that the solution is correct.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that use letters to represent unknown quantities, often referred to as variables, and express relationships between those quantities using mathematical operations. For instance, in our exercise, "The quotient of 49 and a number is 7," the word "quotient" signals division, with the number to be found denoted as \( x \). Hence, the equation would be expressed as \( \frac{49}{x} = 7 \).

Here’s a basic guide to forming algebraic equations:

• Identify the unknown variables, often with symbols like \( x \), \( y \), etc.
• Determine the mathematical operations implied by the wording, such as addition, subtraction, multiplication, or division.
• Translate the verbal statement into an algebraic equation accurately reflecting the stated relationship.

Understanding how to translate word problems into algebraic equations is crucial in solving many math problems. This allows you to simplify real-world problems into manageable mathematical expressions.

Quotient in Math

The term "quotient" is frequently found in mathematical problems and refers to the result of division between two numbers. In the phrase, "the quotient of 49 and a number is 7," it refers to what you get when you divide the number 49 by an unknown value, resulting in 7. This can be mathematically expressed as an equation: \( \frac{49}{x} = 7 \).

To solve for \( x \), which represents the unknown number, follow these steps:

• Multiply both sides by \( x \) to eliminate the fraction, resulting in \( 49 = 7x \).
• Isolate \( x \) by dividing both sides by 7, giving \( x = 7 \).

By understanding quotient, you can accurately interpret and set up equations, leading to correct solutions.

Checking Solutions in Algebra

Verifying your solution is a vital step in solving algebraic equations. After you find a potential solution, it's important to substitute it back into the original equation to ensure it holds true. In our example, upon solving \( \frac{49}{x} = 7 \) and finding \( x = 7 \), we substitute back to check:

• Plug \( x = 7 \) into the original equation: \( \frac{49}{7} = 7 \).
• Both sides simplify to 7, confirming that the solution is correct.

This process of checking reinforces understanding and avoids errors. Ensuring your solution satisfies the original equation is a best practice in algebra, providing confidence in your work.