Problem 8 - Intermediate Algebra

Question

Fill in the blanks. If \(a=b,\) then \(c a=\square\) and \(\frac{a}{c}=\frac{b}{\square} .\) Multiplying (or dividing) both sides of an equation by the _____ nonzero number does not change the solution.

Step-by-Step Solution

Answer

The filled equation is: If \(a=b\), then \(ca=cb\) and \(\frac{a}{c}=\frac{b}{c}\). The word to complete the sentence is 'same'.

Step 1: Understanding the First Equation

We start with the equation \(a = b\). If we multiply both sides of this equation by the same number, the equality is still valid. For the first blank, we are multiplying both sides by \(c\), therefore the right side becomes \(c \cdot b\). So, the first blank should be filled with \(cb\).

Step 2: Understanding the Second Equation

Given \(\frac{a}{c} = \frac{b}{\square}\), we are tasked with finding the correct number that makes the equation true. Since \(a = b\), dividing both sides by \(c\) should keep the equation balanced. Thus, the expression on the right side should match the one on the left, \(\frac{a}{c} = \frac{b}{c}\). So, the second blank should be filled with \(c\).

Step 3: Understanding Multiplying or Dividing by the Same Number

In mathematics, multiplying or dividing both sides of an equation by the same nonzero number does not change the solution. This is because nonzero numbers maintain the balance of the equation. Therefore, the blank here should be filled with the word 'same'.

Key Concepts

Multiplication Property of Equality

The multiplication property of equality is a fundamental idea in algebra that allows us to maintain the equality between two expressions when both sides of an equation are multiplied by the same nonzero number. Consider starting with an equation like \(a = b\). If you multiply both sides by a number, say \(c\), the equation will transform into \(ca = cb\). This operation doesn't change the balance or the solution of the equation because the same number \(c\) is affecting both sides equally.

• It's important that the multiplier is the same for both sides.
• Multiplying by zero would lead to losing information, since \(ca = cb\) would both be zero if \(c = 0\), which isn't helpful.
• Thus, the number \(c\) must be nonzero to keep the equation valid and meaningful.

Knowing this property can help solve equations by isolating variables or converting setups into simpler forms, maintaining their equivalences.

Division Property of Equality

Just as with multiplication, dividing both sides of an equation by the same nonzero number will keep the original equality intact. If we have \(a = b\) and divide each side by \(c\), the result is \(\frac{a}{c} = \frac{b}{c}\).

• This is particularly useful when simplifying equations or clearing fractions.
• However, you must ensure \(c\) is nonzero to prevent division by zero, which is undefined in mathematics.
• By dividing by a number on both sides, we are essentially scaling down the sides evenly, which maintains their equality.

This property gives a powerful way to simplify and solve equations by systematically reducing complexity, and by maintaining the truth of the equation across each step.

Maintaining Equation Balance

Maintaining equation balance is the essence of both the multiplication and division properties of equality. An equation is like a perfectly balanced scale, where both sides need to remain equal. Whatever is done to one side, must be done to the other to keep the balance.

• Performing operations like addition, subtraction, multiplication or division by the same quantity on both sides keeps the equation valid.
• Balancing is critical in solving equations since this principle underlies the ability to solve for unknowns systematically.
• For equations, think of balance as performing equal actions that don't disrupt or destroy the equilibrium established by the original statement.

This concept is pivotal in algebra where keeping operations equal ensures that the solutions derived are correct and reflect the original conditions posed by the problem.