Problem 3 - Algebra and Trigonometry

Question

True or false? (a) Adding the same number to each side of an equation always gives an equivalent equation. (b) Multiplying each side of an equation by the same number always gives an equivalent equation. (c) Squaring each side of an equation always gives an equivalent equation.

Step-by-Step Solution

Answer

(a) True, (b) True, (c) False.

Step 1: Analyze Statement (a)

Statement (a) is about adding the same number to each side of an equation. The property of equality states that adding the same number to both sides of an equation keeps the equation balanced. This operation does not change the solution set, so the statement is true.

Step 2: Analyze Statement (b)

Statement (b) discusses multiplying each side by the same number. The multiplication property of equality holds true as long as the number is not zero. Multiplying both sides of an equation by the same nonzero number maintains its equivalence, so this statement is true.

Step 3: Analyze Statement (c)

Statement (c) involves squaring both sides of an equation. Squaring each side can introduce extraneous solutions, making the original and resulting equations potentially unequal. For example, if \( x = 1 \), then \( x^2 = 1 \) and \((-1)^2 = 1\) could falsely appear equivalent. Therefore, this statement is false.

Key Concepts

Properties of Equality

When solving equations, the properties of equality are fundamental rules used to maintain balance. These properties ensure that whatever you do to one side of the equation, doing the same to the other side will keep the equation equivalent. The basic aim is to alter equations without changing their solution set.
Some common properties include:

• **Addition Property**: Adding a specific number to both sides.
• **Subtraction Property**: Subtracting a specific number from both sides.
• **Multiplication Property**: Multiplying both sides by a nonzero number.
• **Division Property**: Dividing both sides by a nonzero number.

These rules are about preserving the balance of the equation, thereby assuring any operations performed are valid and the original equation's truth is retained.

Adding to Both Sides

Adding the same number to both sides of an equation is one of the most direct applications of the properties of equality. If you have an equation such as \(x + 3 = 7\), you can decide to add (-3) to both sides to maintain balance and solve for \(x\).
Here's why it works:

• By adding the same value, you don't alter the essential relationship between the quantities on either side.
• This process aids in isolating the variable, often to reveal its value.

So, the result remains equivalent. For example, if \(a = b\), then \(a + c = b + c\) for any number \(c\). This practice ensures you have an accurate and unchanged equality.

Multiplying Both Sides

The multiplication property of equality involves multiplying each side by the same number, excluding zero. This operation assists in maintaining the equality while adjusting the equation's coefficients.
For instance, consider the equation \(2x = 8\). By multiplying each side by \(\frac{1}{2}\), you can determine that \(x = 4\).

• This property allows you to simplify or find the solution of an equation without altering the position of the equal sign.
• Ensuring that the number isn't zero is critical, as multiplying by zero would invalidate the properties of equality by reducing everything to zero.

The multiplication property proves invaluable in solving equations, ensuring adjustments are accurate and valid.

Effect of Squaring

Squaring both sides of an equation is not as straightforward as adding or multiplying, primarily because it can introduce extraneous solutions. If you have an equation like \(x = 3\), squaring both sides yields \(x^2 = 9\). It's true that \(\pm 3\) both satisfy the squared equation, even though \(x = -3\) wasn't a solution in the original equation.

Here's what squaring does:

• Introduces new solutions that may not satisfy the original equation.
• Can potentially mask the original relationship between terms.

So, while squaring can sometimes provide solutions, it is critical to check each result against the original equation to confirm its validity.