Problem 100 - Beginning and Intermediate Algebra

Question

Write each of the following as a mathematical expression, and simplify. Subtract \(3 x-5\) from \(2 x-3\).

Step-by-Step Solution

Answer

-x + 2

Step 1 - Write the subtraction as an expression

To subtract the expression \(3x-5\) from \(2x-3\), write it as \( (2x - 3) - (3x - 5) \).

Step 2 - Distribute the negative sign

Distribute the negative sign through the second parenthesis: \( 2x - 3 - 3x + 5 \).

Step 3 - Combine like terms

Combine like terms: \(2x - 3x - 3 + 5 \). This simplifies to \(-x + 2 \).

Key Concepts

Subtraction of Expressions

In algebra, subtracting one expression from another requires careful handling of negative signs. When we are given an instruction such as 'Subtract \(3x-5\) from \(2x-3\)', we need to translate this into a mathematical expression.
Start by writing the subtraction operation as: \( (2x - 3) - (3x - 5) \). Notice how we use parentheses to clearly distinguish between the two expressions.
This initial step lays the groundwork for all subsequent algebraic manipulations.

Distribution

Distribution involves multiplying a term outside a parenthesis across each term inside the parenthesis. In our problem, we need to distribute a negative sign across the expression inside the parentheses.
Look at \( (2x - 3) - (3x - 5) \). Here, we distribute the negative sign across each term in \( (3x - 5) \):
\[ 2x - 3 - 3x + 5 \]
This is because subtracting is equivalent to adding a negative. So, \( - (3x - 5) \) turns into \( -3x + 5 \). The parenthesis are removed once distribution is complete. This step is crucial for avoiding common mistakes with signs.

Combining Like Terms

Combining like terms is all about simplifying the expression by adding or subtracting terms that have the same variable. After distributing, our expression is:
\[ 2x - 3 - 3x + 5 \]
We then look for terms that share the same variable and combine them. Here, \( 2x \) and \( -3x \) are like terms, and \( -3 \) and \( 5 \) are constants that can be combined:
\[ 2x - 3x - 3 + 5 \]
Combining these, we get:
\[ -x + 2 \]
Combining like terms simplifies the expression, making it easier to understand and work with. In our final simplified form, \( -x + 2 \), we see that this is much simpler than the original expression.