Problem 10 - Algebra 1: Concepts and Skills

Question

Add or subtract, then factor and simplify. $$ \frac{2 y+3}{y^{2}-4 y}-\frac{-y+15}{y^{2}-4 y} $$

Step-by-Step Solution

Answer

The simplified version of the given expression is \(\frac{3}{y}\).

Step 1: Recognize the Common Denominator

Notice that the denominators of the two fractions are identical, \(y^{2}-4 y\). This means we can combine the two fractions.

Step 2: Subtracting the Numerators

Subtract the numerators of the fractions: \((2y + 3) - (-y +15)\), which simplifies to \(2y + y + 3 - 15 = 3y - 12\).

Step 3: Writing the New Fraction

Now we can write the new fraction as: \(\frac{3y - 12}{y^{2} - 4y}\).

Step 4: Factoring

Factor both the numerator and the denominator. This gives: \(\frac{3(y - 4)}{y(y - 4)}\).

Step 5: Simplifying the Fraction

Finally, we can cancel out the common factor of \(y - 4\) in the numerator and denominator. The final solution is therefore \(\frac{3}{y}\).

Key Concepts

Combining Like Terms

Understanding how to combine like terms is fundamental to simplifying algebraic expressions. Like terms are terms that have the exact same variable raised to the same power. For instance, in the expression \(2y + 3 - (-y + 15)\), \(2y\) and \(y\) are like terms because they both contain the variable \(y\) raised to the first power.

To combine them, you simply add or subtract their coefficients. For the positive \(y\), its coefficient is implicitly 1. Thus, \(2y\) and \(1y\) combine to \(3y\). In the given exercise, we also subtract the constants: \(3 - 15\) combines to \( -12\). This step simplifies the numerator to \(3y - 12\), setting the stage for the further simplification of the fraction.

Factoring Polynomials

Factoring polynomials involves writing them as a product of their factors. This can significantly ease the process of simplifying algebraic fractions. A key technique is to look for a common factor in all terms of the polynomial, known as the greatest common factor (GCF). In our example, the numerator \(3y - 12\) has a GCF of 3, which when factored out leaves us with \(3(y - 4)\).

Similarly, for the denominator \(y^2 - 4y\), the GCF is \(y\), leaving us with \(y(y - 4)\). Factoring exposes common factors in the numerator and denominator that can be simplified, which leads directly to the next step—simplification, an essential move to arrive at the most reduced form of the algebraic fraction.

Common Denominator

Finding a common denominator is crucial for adding or subtracting fractions, including algebraic fractions. The denominator represents the 'whole' in terms of which the fractions are expressed. If fractions have different denominators, they are like apples and oranges—cannot directly be combined.

In algebraic fractions, if the denominators are the same or can be made the same, they are, in essence, already speaking the same 'language'. It's akin to ensuring everyone at a meeting talks about the same topic to reach a meaningful conclusion. In our exercise, \(y^2 - 4y\) serves as a common denominator, which enables direct subtraction of the numerators. This shared denominator reflects a shared basis for the fractions, so they can be straightforwardly subtracted or added to simplify the expression.