Problem 80 - Algebra A Combined Function

Question

Fill in the squares so that each is a true statement. $$ 9 y^{7}+3 y^{\square}=12 y^{7} $$

Step-by-Step Solution

Answer

Replace the square with 7.

Step 1: Understand the Equation

The given equation is \( 9y^{7} + 3y^{\square} = 12y^{7} \). Our goal is to find an integer that will replace the square such that the equation holds true. Notice that the terms are made of powers of \(y\).

Step 2: Simplify the Equation

To simplify \( 9y^{7} + 3y^{\square} = 12y^{7} \), subtract \(9y^{7}\) from both sides:\[3y^{\square} = 12y^{7} - 9y^{7}\]This further simplifies to:\[3y^{\square} = 3y^{7}\]

Step 3: Solve for the Unknown Exponent

Since we now have \(3y^{\square} = 3y^{7}\), divide both sides by 3:\[y^{\square} = y^{7}\]This implies that the exponent \(\square\) must be equal to 7. Thus, the missing exponent is 7.

Key Concepts

Simplifying Equations

Simplifying equations is a fundamental skill in algebra that helps us to unravel complex expressions and solve for unknown values. In the given exercise, we need to simplify the equation to make it easier to solve. Start by looking for like terms. In our example, we have terms like \(9y^7\) and \(12y^7\), which are both multiples of \(y^7\). This means they can be combined or simplified together. To simplify the given equation \(9y^{7} + 3y^{\square} = 12y^{7}\):

• First, subtract \(9y^7\) from both sides, which allows us to isolate the unknown term on one side.
• This leads to the equation: \(3y^{\square} = 3y^7\).

This simplification process reduces the complexity of the problem, allowing us to focus on solving for the unknown exponent. Remember, whenever you're simplifying equations, always keep the equation balanced by doing the same operation on both sides.

Solving Exponents

Exponents represent repeated multiplication of a base number. In this exercise, we encounter expressions like \(y^7\) and need to find an unknown exponent. Solving an equation with variables in the exponent often involves equating the exponents, provided the base is the same.Here’s a quick way to handle exponents in this context:

• When we have \(3y^{\square} = 3y^7\), notice that both sides have the same base, \(y\).
• We can divide both sides by 3, simplifying the equation to \(y^{\square} = y^7\).
• Once the bases are the same and isolated, compare the exponents directly: \(\square = 7\).

Solving exponents becomes more straightforward when the equation involves a common base, as it allows you to isolate the exponents. This fundamental approach enables you to tackle more complex problems involving powers and roots.

Polynomial Expressions

Polynomial expressions consist of terms that involve variables raised to different powers, each multiplied by a coefficient. Recognizing polynomials is essential in algebra, especially when simplifying or solving equations involving them.For example, the expression \(9y^7 + 3y^{\square}\):

• The term \(9y^7\) is a part of the polynomial, where 9 is the coefficient and 7 is the exponent.
• Similarly, \(3y^{\square}\) represents another polynomial term with an unknown exponent.

To manage polynomial expressions:

• Always identify and combine like terms, which are terms with the same variable raised to the same power.
• Simplify wherever possible by performing arithmetic operations on coefficients.
• Use principles of like terms and distributive properties effectively.

Handling polynomial expressions well can simplify the process of solving equations and allows you to focus directly on finding unknown values and completing algebraic manipulations. With these approaches, you’ll be better equipped to dissect and solve equations with multiple terms and variables.