Problem 31 - Elementary Algebra

Question

Subtract \(5 x-1\) from \(2 x-3\).

Step-by-Step Solution

Answer

The simplified expression is \(-3x - 2\).

Step 1: Understand the Problem

We need to subtract the expression \(5x - 1\) from the expression \(2x - 3\). This involves finding the difference between these two algebraic expressions.

Step 2: Set Up the Subtraction

To subtract \(5x - 1\) from \(2x - 3\), we write it as \((2x - 3) - (5x - 1)\). Be sure to distribute the negative sign to each term in the expression \(5x - 1\).

Step 3: Distribute the Negative Sign

Distribute the negative sign across the terms in the equation: \((2x - 3) - 5x + 1\). The expression \(5x - 1\) becomes \(-5x + 1\) when subtracted.

Step 4: Combine Like Terms

Combine like terms in the expression \(2x - 3 - 5x + 1\). First, combine the \(x\) terms: \(2x - 5x = -3x\). Then, combine the constant terms: \(-3 + 1 = -2\).

Step 5: Write the Final Simplified Expression

The result of combining the terms is \(-3x - 2\). This is the simplified expression after subtracting \(5x - 1\) from \(2x - 3\).

Key Concepts

Subtraction of Expressions

When we perform subtraction with algebraic expressions, we're finding the difference between two or more terms. It's important to remember the order: subtract the second expression from the first one given. Let's look at our example: subtracting \(5x - 1\) from \(2x - 3\). This means we write it as \((2x - 3) - (5x - 1)\).

We follow these steps:

• Write down the first expression, \(2x - 3\).
• Subtract the entire second expression \(5x - 1\), by placing it in parentheses and applying a negative sign in front of it.

This setup helps avoid errors as we perform the subtraction correctly.

Combining Like Terms

When simplifying expressions, we need to combine like terms. Like terms are terms that have the same variables raised to the same power.

For example:

• In \(2x - 3 - 5x + 1\), the like terms are \(2x\) and \(-5x\), both having the variable \(x\).
• The constants \(-3\) and \(+1\) are also like terms.

Combine the \(x\) terms: \(2x - 5x\) simplifies to \(-3x\).
Combine the constants: \(-3 + 1\) becomes \(-2\).
These steps result in the simplified expression \(-3x - 2\). Combining like terms helps make expressions easier to understand and solve.

Distributive Property

The distributive property is key in algebra, especially when subtracting expressions. It tells us to multiply a number or variable outside the parentheses by all terms inside the parentheses.

In our example, we distribute the negative sign to \(5x - 1\):

• First, multiply \(-1\) by \(5x\) to get \(-5x\).
• Next, multiply \(-1\) by \(-1\) to get \(+1\).

Thus, \((5x - 1)\) becomes \(-5x + 1\) when subtracted.
This method ensures that the subtraction affects each component of the expression accurately. Mastery of the distributive property is essential for effective algebraic manipulation.