Problem 61 - Algebra A Combined Function

Question

Perform each indicated operation. Don't forget to simplify if possible. Subtract \(7 x+1\) from \(3 x-8\)

Step-by-Step Solution

Answer

The result of the subtraction is \(-4x - 9\).

Step 1: Write the Expression

We need to perform the subtraction of two expressions. Start by writing down the two expressions involved in the operation. These are:- The first expression: \(3x - 8\)- The second expression: \(7x + 1\)

Step 2: Subtract the Expressions

To carry out the subtraction, you need to subtract the second expression from the first. This can be represented as:\[(3x - 8) - (7x + 1)\]Make sure to distribute the subtraction sign across the second expression.

Step 3: Distribute the Negative Sign

Distribute the negative sign over the second expression:\[3x - 8 - 7x - 1\]This changes the operation to subtract \(7x\) and subtract \(1\).

Step 4: Combine Like Terms

Now, combine the like terms:- Combine the \(x\) terms: \(3x - 7x = -4x\)- Combine the constant terms: \(-8 - 1 = -9\)Putting it together, you get:\(-4x - 9\)

Step 5: Simplify the Expression

The simplified form of the expression after performing the subtraction is:\(-4x - 9\)There are no like terms to combine further, so this is the final answer.

Key Concepts

Subtraction of Expressions

Subtracting expressions in algebra is a foundational concept that helps in understanding further algebraic operations. When you subtract one expression from another, you need to focus on the positioning of terms and the impact of the subtraction operation on each term.

In the example given, the task is to subtract \(7x + 1\) from \(3x - 8\). Notice that the expression being subtracted needs a parenthesis around it due to the subtraction sign:

• First Expression: \(3x - 8\)
• Second Expression (to subtract): \(7x + 1\)

Place the first expression and then subtract the entire second expression: \((3x - 8) - (7x + 1)\). This sets the stage for distributing and simplifying.

Distributing Negative Signs

Distributing a negative sign is critical when subtracting one expression from another.

Think of the negative sign as changing the sign of each element in the expression that follows it. In our example, we have \((3x - 8) - (7x + 1)\). To simplify this, you need to distribute the negative sign to every term within the parentheses:

• The \(7x\) becomes \(-7x\).
• The \(+1\) becomes \(-1\).

Once the negative sign is distributed, the expression transforms into: \(3x - 8 - 7x - 1\). Now, you're ready to combine like terms for further simplification.

Combining Like Terms

Combining like terms is the final step in simplifying expressions, ensuring you group similar kinds of terms. Like terms have the same variable raised to the same power. In our case, this involves grouping terms with \(x\) together and constants together.

After distributing the negative sign, we have the expression: \(3x - 8 - 7x - 1\).

• Combine the \(x\) terms: \(3x - 7x = -4x\).
• Combine the constant terms: \(-8 - 1 = -9\).

Therefore, the simplified form of the original subtraction expression is \(-4x - 9\). This shows that careful attention to like terms ensures you arrive at a correct and simplified final result.