Problem 20 - Algebra 1: Concepts and Skills

Question

Find the missing numerator. $$ \frac{x-9}{2 x+3}=\frac{?}{x(2 x+3)} $$

Step-by-Step Solution

Answer

The missing numerator is \(x^2 - 9x\).

Step 1: Express the Equation

The question gives the equation \(\frac{x-9}{2x+3} = \frac{?}{x(2x+3)}\).

Step 2: Identify the Missing Numerator

Express the left hand side of the equation entirely in terms that appear in the right hand side. In this case, comparing the denominators we can see that we have factored out an x from the denominator - so this x must multiply the numerator. The missing numerator is \(x * (x - 9)\). Apply the distributive property of multiplication over subtraction to get \(x^2 - 9x\). So the missing numerator is \(x^2 - 9x\).

Step 3: Simplify the Equation

We simplified the equation to \(\frac{x^2 - 9x}{x(2x + 3)}\) thereby finding the missing numerator.

Key Concepts

Understanding Rational Expressions

Rational expressions are like fractions but with polynomials in the numerator or denominator instead of regular numbers. In simpler words, a rational expression is one polynomial divided by another. Think of it as any algebraic expression that has a fraction form where both the top and bottom have variables involved, not just constants.
Working with rational expressions involves performing operations like addition, subtraction, multiplication, and division, just like with regular fractions. But, these operations become a bit more complex due to the presence of variables.

• When you add or subtract rational expressions, you need a common denominator, just like with numeric fractions.
• When multiplying or dividing them, it's crucial to factor and reduce wherever possible to simplify your work.

In the given exercise, both sides of the equation are rational expressions where maintaining equivalency and solving for the unknown remains the central task.

Applying the Distributive Property

The distributive property is a fundamental building block in algebra that helps us simplify expressions and solve equations more easily. It states that for any three numbers or algebraic expressions, you can multiply a sum by multiplying each addend individually and then add the products together. Mathematically, this is written as: \[ a(b+c) = ab + ac \]For example, if you have 3(4+5), you can distribute the 3 to get 3*4 + 3*5, which simplifies to 12 + 15 = 27. In the context of algebra, particularly with polynomials, we often use the distributive property to expand expressions and to simplify terms.
In the exercise we are looking at, we used the distributive property to expand the expression \( x(x-9) \), which results in \( x^2 - 9x \). This application helps us find the missing numerator by expanding and simplifying the product of \( x \) and \( x-9 \).

Solving Equations in Algebra

Equation solving is the process of finding the value of the variables that make the equation true. This typically involves isolating the variable on one side of the equation and making adjustments to each side as necessary to maintain balance. Much like balancing a scale!Some key strategies in equation solving include:

• Applying operations evenly to both sides of an equation to maintain equality (like adding, subtracting, multiplying, or dividing).
• Using properties of equality and inverse operations to simplify and solve.
• Factoring, expanding, or other algebraic techniques to isolate the variable.

In rational expressions, particularly, it’s important to understand the role of common denominators. As with common fractions, mismatched denominators can hinder our ability to solve directly until they are addressed. In the given problem, understanding how denominators relate helped us identify the necessity to include an additional term (the missing factor of \( x \)) on one side.