Problem 2 - Prealgebra

Question

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$\frac{1}{3} x=7$$

Step-by-Step Solution

Answer

The solution is \( x = 21 \).

Step 1: Understand the Equation

The given equation is \( \frac{1}{3} x = 7 \). This means that one-third of \( x \) is equal to 7.

Step 2: Identify the Multiplication Property of Equality

To solve for \( x \), we can use the multiplication property of equality, which states that if we multiply both sides of an equation by the same number, the two sides remain equal.

Step 3: Multiply Both Sides by 3

To eliminate the fraction, multiply both sides of the equation by 3: \( 3 \times \frac{1}{3} x = 3 \times 7 \).

Step 4: Simplify the Equation

When you multiply \( 3 \times \frac{1}{3} \), the result is \( 1 \), thus simplifying to \( x = 21 \).

Step 5: Verify the Solution

Substitute \( x = 21 \) back into the original equation: \( \frac{1}{3} \times 21 = 7 \). Simplifying gives \( 7 = 7 \), confirming the solution is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics that allows us to find the value of an unknown variable. When we have an equation, we want both sides to be balanced or equal to each other. The process involves manipulating the equation until the unknown variable is isolated on one side. This manipulation must maintain the equality, meaning any operation done on one side must be applied to the other side as well. Here is a step-by-step method:

• Identify the equation and understand what it means. For example, in \( \frac{1}{3}x = 7 \), one-third of \( x \) equals 7.
• Decide on a suitable method to isolate the variable, like using the multiplication property of equality.
• Perform the required operations with care to maintain balance in the equation.
• Simplify the result to find the value of the unknown variable.
• Verify the solution by substituting it back into the original equation to ensure both sides equal.

Mastering solving equations is crucial, as it is the foundation of more advanced topics in mathematics, such as algebra and calculus.

Fractions in Equations

Fractions can often complicate equations, but understanding them is crucial for solving problems efficiently. When an equation contains a fraction, such as \( \frac{1}{3}x = 7 \), the goal is to eliminate the fraction to simplify the equation. Here are some tips:

• Identify the fraction and determine how it affects the variable.
• Use multiplication to remove the fraction. In this context, multiplying both sides by the denominator, which is 3, effectively cancels out the fraction:
  \[ 3 \times \frac{1}{3}x = 3 \times 7 \].
• After multiplying, simplify to find the variable, resulting in \( x = 21 \).
  

By converting fractional equations into simpler forms, you can more easily see the solutions. It's important to understand that fractions don't change the fundamental equality of the equation; they just add a step in the simplification process.

Mathematical Proofs

A mathematical proof is a logical argument that uses previously established statements to demonstrate the truth of a statement. Proving equations, like verifying a solution, ensures the correctness of an answer. Here's how to incorporate proofs in solving equations:

• Begin with the original equation. For example, \( \frac{1}{3}x = 7 \) is our starting point.
• Solve the equation using valid mathematical steps, such as multiplication or addition.
• After finding a solution such as \( x = 21 \), plug it back into the original equation to check:
  \[ \frac{1}{3} \times 21 = 7 \].
• Observe the result. If both sides are equal, the solution is verified.

Using mathematical proofs in this way not only aids in understanding the process but also boosts confidence in the solution's accuracy. It is a disciplined approach to ensure that each step in an equation is valid and reliable.