Problem 12 - Elementary Algebra

Question

Simplify. $$ -3 y+25 y-10+y+75 y-10-3 y+45 y-10 $$

Step-by-Step Solution

Answer

The simplified expression is \(140y - 30\).

Step 1: Identify Like Terms

Identify and group the like terms in the expression. The like terms are those that have the same variable part. In this expression, the variable part is 'y'. The constant terms are the numbers without 'y'. So, the like terms here are: - Terms with 'y': - -3y, 25y, y (which is 1y), 75y, -3y, and 45y - Constant terms: - -10, -10, and -10

Step 2: Combine Like Terms with 'y'

Combine the coefficients of the terms containing 'y':\(-3 + 25 + 1 + 75 - 3 + 45 = 140\)So, the terms with 'y' simplify to:\(140y\)

Step 3: Combine Constant Terms

Combine the constant terms:\(-10 - 10 - 10 = -30\)The constant terms simplify to:\(-30\)

Step 4: Write the Simplified Expression

Combine the results from Step 2 and Step 3 to write the simplified expression:The expression simplifies to:\(140y - 30\)

Key Concepts

Like Terms

When working with algebraic expressions, you'll often encounter the term "like terms." These are terms in an algebraic expression that have the same variable component raised to the same power, which means they can be combined. For example, in the expression \(-3y + 25y - 10 + y + 75y - 10 - 3y + 45y - 10\), the like terms are those that involve the variable 'y'. Let's break down the expression:

• Terms with 'y': \(-3y\), \(25y\), \(1y\), \(75y\), \(-3y\), and \(45y\)
• Constant terms: \(-10, -10, -10\)

Notice how all 'y' terms share the same variable part 'y', indicating these are indeed like terms. Recognizing and grouping like terms help simplify complicated algebraic expressions, making them much easier to work with.

Combining Coefficients

Once you've identified like terms in an expression, the next step is to combine their coefficients. Coefficients are the numerical parts of the terms that accompany the variables. In the case of the expression we're simplifying, the coefficients of terms with 'y' are:

• \(-3\) from \(-3y\)
• \(25\) from \(25y\)
• \(1\) from \(y\)
• \(75\) from \(75y\)
• \(-3\) from \(-3y\) again
• \(45\) from \(45y\)

To combine these coefficients, sum them up: \(-3 + 25 + 1 + 75 - 3 + 45 = 140\). The resulting combined term becomes \(140y\). By combining like terms through their coefficients, you effectively reduce the complexity of the expression, simplifying it for easier calculations later on.

Constant Terms

Constant terms in an algebraic expression are those that don't contain any variables, just numbers. In the exercise expression, the constant terms are \(-10, -10, and -10\). These can also be simplified by simply adding, or in this case, combining the negative values together.Let's see how to deal with these:

• Add the constant terms: \(-10 - 10 - 10\). When combined, these simplify to \(-30\).

Constant terms are straightforward because they don’t change—there's no variable affecting them. They’re a key part of writing the final simplified expression, which combines both the variable terms (like terms) and constant terms to give the simplest form of the expression. The final expression in our exercise becomes \(140y - 30\), uniting both the simplified like terms and constants.