Question

Yes or No? If No, give a reason.

1. When you add the same number to each side of an equation, do you always get an equivalent equation
2. When you multiply each side of an equation by the same nonzero number, do you always get an equivalent equation?
3. When you square each side of an equation, do you always get an equivalent equation?

Short answer

1. Yes, when you add the same number to each side of an equation, you always get an equivalent equation.
2. Yes, when you multiply each side of an equation by the same nonzero number, you always get an equivalent equation.
3. Yes, when you square each side of an equation, you always get an equivalent equation.

Step-by-step solution

Part a. Step 1. Apply the concept of equations.

An equation is a statement that two mathematical expressions are equal.

Part a. Step 2. Apply the concept of equivalent equations.

Two equations with exactly the same solutions are called equivalent equations.

Part a. Step 3. Analyze the given situation.

When you add the same number to each side of an equation, we have:

$A=B\iff A+C=B+C$

So, when you add the same number to each side of an equation, you always get an equivalent equation.

Part b. Step 1. Apply the concept of equations.

An equation is a statement that two mathematical expressions are equal.

Part b. Step 2. Apply the concept of equivalent equations.

Two equations with exactly the same solutions are called equivalent equations.

Part b. Step 3. Analyze the given situation.

When you multiply each side of an equation by the same nonzero number, we have:

$A=B\iff AC=BC$

So, when you multiply each side of an equation by the same nonzero number, you always get an equivalent equation.

Part c. Step 1. Apply the concept of equations.

An equation is a statement that two mathematical expressions are equal.

Part c. Step 2. Apply the concept of equivalent equations.

Two equations with exactly the same solutions are called equivalent equations.

Part c. Step 3. Analyze the given situation.

When you square each side of an equation, we have:

$A=B\iff A^2=B^2$

So, when you square each side of an equation, you always get an equivalent equation.