Question

If \(A=2 x-(y-2 c)\) and \(B=(2 x-y)-2 c,\) then \(A-B=\) a. \(-2 y\) b. \(-4 c\) c. \(-0\) d. \(-2 y\) e. \(-4 c\)

Short answer

c. 0

Step-by-step solution

Substitute the values of A and B into A-B

To start, let’s input the given values of \(A\) and \(B\) into the expression \(A-B\). This gives us \((2x - (y - 2c)) - ((2x - y) - 2c)\)

Carry out the subtraction operation

Now, carefully subtract the expressions within the brackets. Recall rules of changing signs when subtracting expressions. This gives us \(2x - y + 2c - 2x + y - 2c\)

Simplify the resulting expression

Upon simplifying, we can clearly see that every term cancels each other out (positive 2x cancels negative 2x, negative y cancels positive y, positive 2c cancels negative 2c), leaving us with \(0\)

Key concepts

Subtraction of Expressions

When you come across subtracting two algebraic expressions like the given expressions for \(A\) and \(B\), the key is to carefully deal with the subtraction sign, especially when it affects terms in parentheses. This might seem tricky at first because subtracting expressions is a bit different from subtracting numbers. Here are some helpful steps:

• Write down each expression clearly.
• Place each expression within its own pair of parentheses.
• Apply the subtraction operation to the expressions.

In our problem, the subtraction had us handle \((2x - (y - 2c)) - ((2x - y) - 2c)\). Each element after the minus sign should be viewed as if it is being multiplied by negative one. This means anything in the second parenthesis will change signs. Remember these rules will help in successfully handling any future subtraction of algebraic expressions!
Be meticulous with signs when dealing with subtraction in order to avoid any potential errors.

Expression Simplification

An important aspect of solving algebraic problems is expression simplification. Simplification makes expressions easier to understand and compare. Once you correctly apply the subtraction rules, as in our exercise, the next step is to simplify the resulting expression.

• Combine like terms: Terms that have the exact same variables are added or subtracted together.
• Arrange the terms neatly, grouping similar terms together.

In the example provided, the expression \(2x - y + 2c - 2x + y - 2c\) simplifies straightforwardly because every term has an equal and opposite counterpart. This leads directly to our simplified solution, \(0\).
Simplifying expressions allows for clearer insight into what an expression actually represents, aiding in further evaluation or substitution as needed.

Canceling Terms

Canceling terms is a valuable technique during simplification and helps you spot how terms can neutralize each other. In algebraic expressions, terms can be canceled out if they are the same but have opposite signs. Here’s a simple guide:

• Look for pairs of terms with opposite signs.
• Observe if one term can "terminate" or cancel the other, reducing complexity.

This process explains why the algebraic expression \(2x - y + 2c - 2x + y - 2c\) collapses to \(0\). Each term in the expression had an exact opposite, allowing them to perfectly cancel each other out. When terms are successfully canceled, you are left with either a simpler expression or, as in our case, nothing at all!
Learning how terms cancel each other not only simplifies calculations but also helps in better understanding the essence of algebraic manipulations.