Problem 22 - Elementary Algebra

Question

Simplify. $$ 1 x-12 y $$

Step-by-Step Solution

Answer

The simplified expression is \( x - 12y \).

Step 1: Identify the Terms

The expression given is \( 1x - 12y \). It consists of two terms: \( 1x \) and \( -12y \).

Step 2: Simplify the Coefficient

Review the coefficient of the \( x \) term. The coefficient of \( x \) is 1, which implies we can simplify \( 1x \) to \( x \).

Step 3: Rewrite the Expression

After simplifying the \( x \) term, rewrite the expression as \( x - 12y \). This expression is already in its simplest form because no like terms can be combined.

Key Concepts

Algebraic Terms

Algebraic terms are the building blocks of an expression like the one we're working with, which is \(1x - 12y\). Each term in the expression consists of:

• A coefficient (which we'll dive into in the next section).
• A variable (letters like \(x\) and \(y\) that represent unknown values).
• Sometimes powers (exponents), although not present in this specific example.

In our example, \(1x\) and \(-12y\) are the two algebraic terms. Each one stands alone because they aren't like terms (no two terms have the same variable raised to the same power). It's important to note the '-' sign in front of 12, which makes the second term negative. The goal when simplifying an expression is to combine like terms. Here, since the terms involve different variables, no further combination is possible.

Coefficients

When breaking down an algebraic expression, the term 'coefficient' crops up frequently. In the given exercise, \(1x - 12y\), coefficients are numerical values that multiply a variable.

• In \(1x\), the coefficient is 1. It shows that the variable \(x\) is scaled by a factor of 1.
• In \(-12y\), the coefficient is -12, signifying that \(y\) is scaled by a factor of -12.

You can think of coefficients as weights of each variable. They indicate how many times a term's variable should be counted. A coefficient of 1 or -1 is often simplified by removing the '1' or '-1' itself, making the expression cleaner and easier to comprehend.

Simplification Steps

Simplification involves reducing an expression to its most basic form while maintaining its value. In our case, we simplify \(1x - 12y\) by closely looking at each algebraic term.1. **Identify the Terms:** Look at each part of the expression. Here, we have \(1x\) and \(-12y\).2. **Simplify Coefficients:** Specifically, focus on the coefficient of \(1x\). Simplifying 1 by removing it leaves us with \(x\).3. **Rewrite the Expression:** After making those changes, you rewrite the expression as \(x - 12y\), which is now in its simplest form. Since there's no other like term to combine, we conclude the simplification process.
Applying a systematic approach allows us to consistently reach the simplest expression possible. Each step is straightforward, making it easier for learners to remember and apply to more complex problems down the road.