Problem 29 - Algebra 2 and Trigonometry

Question

Solve for the variable and check. Each solution is an integer. \(y(y+2)-y(y-2)=20-y\)

Step-by-Step Solution

Answer

The solution is \( y = 4 \).

Step 1: Expand the Expression

Start by expanding the left hand side of the equation. The given equation is:\[ y(y+2) - y(y-2) = 20 - y \]Expand each term:\[ y^2 + 2y - (y^2 - 2y) = 20 - y \]

Step 2: Simplify the Equation

Combine like terms on the left hand side. Notice the subtraction distributes:\[ y^2 + 2y - y^2 + 2y = 20 - y \]This simplifies to:\[ 4y = 20 - y \]

Step 3: Isolate the Variable

To isolate \(y\), add \(y\) to both sides:\[ 4y + y = 20 \]This gives us:\[ 5y = 20 \]

Step 4: Solve for the Variable

Divide both sides by 5 to solve for \(y\):\[ y = \frac{20}{5} \]Thus, \( y = 4 \).

Step 5: Check the Solution

Substitute \( y = 4 \) back into the original equation to verify:\[ 4(4+2) - 4(4-2) = 20 - 4 \]Calculate each term:\[ 24 - 8 = 16 \]The right side becomes:\[ 20 - 4 = 16 \]Since both sides match, \( y = 4 \) is verified to be correct.

Key Concepts

Expanding Expressions

When we talk about expanding expressions, we’re referring to the process of simplifying or restructuring each part of a mathematical expression so that all components are clearly presented. In the context of equations like \( y(y+2)-y(y-2) \), it's about distributing each term in the multiplication process.
We take the expression and examine each part:

• For \( y(y+2) \), you multiply \( y \) by both \( y \) and \( 2 \) to get \( y^2 + 2y \).
  
• The second expression \( -y(y-2) \) is similar. Here, you distribute \( -y \) across both terms, which results in \( -y^2 + 2y \).

Expanding expressions helps set the stage for further manipulations, leading to a simpler equation to solve. It's a foundational step that makes complex equations easier to handle by breaking them down into more manageable parts.

Combining Like Terms

Combining like terms is another vital step in simplifying equations. Once you have expanded the expression, the next step is to look for terms that are alike—terms that contain the same variable raised to the same power.
In our equation, after expanding, we get:

• \( y^2 + 2y - y^2 + 2y \)

The process involves:

• Identifying like terms: The \( y^2 \) terms cancel each other out in this equation.
• Adding or subtracting coefficients of similar terms: The expression with \( y \)s, adds up to \( 2y + 2y = 4y \).

Combining like terms simplifies the equation by reducing it to a form that is much easier to solve—shorter and less complex, enhancing clarity.

Isolating the Variable

Isolating the variable involves manipulating the equation until the unknown variable is on one side of the equation by itself. This is essential for finding the solution.
In our example, the equation simplifies to \( 4y = 20 - y \). Here’s how you proceed:

• First, add \( y \) to both sides to get all the \( y \) variables on one side, resulting in \( 4y + y = 20 \).
• This simplifies to \( 5y = 20 \).

Now, you have a straightforward equation, \( 5y = 20 \), where only one step remains to solve for \( y \). Achieving this isolation makes it easy to determine the exact value of the variable, a crucial aspect of solving equations.

Checking Solutions

Once you've found a solution, like \( y = 4 \) in this instance, it’s important to verify that it’s correct by substituting it back into the original equation. Checking your solutions confirms its validity.
Substituting back, we rewrite the original equation:

• \( 4(4+2) - 4(4-2) = 20 - 4 \)

Calculate both sides:

• The left side simplifies to \( 24 - 8 = 16 \).
• The right side is \( 20 - 4 = 16 \).

Since both sides are equal, this checks out right! This ensures that \( y = 4 \) satisfies the original equation. Checking solutions is crucial, as it acts as a safety net against any possible errors made during calculations, reinforcing accuracy in your work.