Problem 12 - Fundamentals of Mathematics

Question

When five is added to three times some number, the result is equal to five times the number decreased by seven. What is the number?

Step-by-Step Solution

Answer

The number is 6.

Step 1: Define the variable

Let's define the unknown number as \( x \). This will help us translate the word problem into an equation.

Step 2: Set up the equation from the problem

Based on the problem, "five is added to three times some number," we can express this as \( 3x + 5 \). Similarly, "five times the number decreased by seven" can be expressed as \( 5x - 7 \).

Step 3: Form a mathematical equation

According to the problem, these two expressions are equal. Therefore, we set the equation: \( 3x + 5 = 5x - 7 \).

Step 4: Solve for the variable

Let's solve the equation \( 3x + 5 = 5x - 7 \).1. Subtract \( 3x \) from both sides: \( 5 = 2x - 7 \).2. Add 7 to both sides: \( 12 = 2x \).3. Divide both sides by 2: \( x = 6 \).

Step 5: Verify the solution

Substitute \( x = 6 \) back into the expressions:- For \( 3x + 5 \): \( 3(6) + 5 = 18 + 5 = 23 \).- For \( 5x - 7 \): \( 5(6) - 7 = 30 - 7 = 23 \).Since both expressions are equal, our solution is correct.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra. They're equations where the highest power of the variable is one. In simpler terms, the variable is not squared or raised to any higher power. This makes them straightforward to solve.

In the original exercise, the problem is set in the form of a linear equation. The equation is:

• On one side, we've got three times the unknown number, plus five.
• On the other, we've worked out five times the unknown number minus seven.

Solving a linear equation involves balancing both sides. The equation represents two expressions that are equal to each other. By arranging these equations properly, you can find the value of the unknown variable quite easily. Linear equations like these are key for delving deeper into algebraic concepts.

Variables in Mathematics

Variables are symbols that stand in for unknown numbers. They allow us to build equations to solve real-life problems. In this exercise, we use the variable \( x \) to represent an unknown quantity we're trying to find.

Variables can represent anything: money, objects, time, or in our case, a mysterious number. Naming this unknown quantity helps in constructing equations that mirror real-world situations.

By defining a variable:

• We simplify complex problems into mathematical terms.
• We can use different operations such as addition, subtraction, multiplication, and division to work with the expression involving the variable.
• Ultimately, we solve for the variable by isolating it on one side of the equation.

Understanding variables is crucial as they form the building blocks for learning more advanced algebra concepts.

Problem Solving in Math

Math problems often require a systematic approach to find solutions. In our specific problem, we tackled it using a clear set of steps. This strategy helps in breaking down the problem into manageable parts.

In math problem solving, especially with equations:

• First, identify what you're solving. Here, it was the unknown number.
• Translate the language of the problem into a mathematical form; in this case, this was setting up the equation \( 3x + 5 = 5x - 7 \).
• Next, solve the equation using operations that will isolate the variable on one side.
• Finally, verify your solution by substituting it back into the original expressions to check for accuracy.

This process not only ensures precision but also boosts confidence in tackling a variety of mathematical problems, from basic to complex.