Problem 46 - Algebra for College Students

Question

$$ (y-2)(a+1)=x \quad \text { for } y $$

Step-by-Step Solution

Answer

\(y = \frac{x}{a+1} + 2\)

Step 1: Identify the Equation

We are given the equation \((y-2)(a+1)=x\) and need to solve for \(y\), meaning \(y\) should be isolated on one side.

Step 2: Divide by \((a+1)\)

To eliminate the \((a+1)\) term from the left side of the equation, divide both sides by \((a+1)\). This yields: \(y-2 = \frac{x}{a+1}\), given that \(a+1 eq 0\).

Step 3: Solve for \(y\)

To isolate \(y\), add 2 to both sides of the equation: \(y = \frac{x}{a+1} + 2\).

Step 4: Finalize the Solution

Ensure that all terms are simplified as much as possible. The expression for \(y\) is already in its simplest form: \(y = \frac{x}{a+1} + 2\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions form the building blocks of algebra, consisting of numbers, variables, and arithmetic operations. In the given problem, the expression

• \((y-2)(a+1)=x\)

mixes variables with constants and forms a composite algebraic expression through multiplication. This expression connects several components, where each part like \(y-2\) or \(a+1\) is a smaller expression in itself.
Algebraic expressions are modified following specific rules and operations. Their versatility allows us to describe various mathematical phenomena. To comprehend algebraic expressions, it's important to get comfortable with variables, symbols used to represent unknown or any number. They are essential for creating mathematical models and solving for unknowns in equations. Properly managing algebraic expressions means simplifying them and understanding how to perform operations on them, like addition, subtraction, multiplication, and division.

Equation Manipulation Techniques

Manipulating an equation means performing operations to transform it into a more useful form. Each modification must maintain the equality between both sides.
To manipulate the given equation, \((y-2)(a+1)=x\), start by identifying your goal: isolating \(y\). Each modification hinges on this goal to maintain balance between each equation side and simplify the expression.
Key manipulation strategies include:

• **Distributive Property**: Useful in expanding expressions and simplifying terms.
• **Factoring**: Can simplify complex expressions, providing more manageable forms.
• **Canceling**: Operations that simplify fractions or eliminate terms, as seen when dividing by \((a+1)\).
• **Balancing**: Consistently applying transformations on both sides, keeping the equality intact.

By methodically applying these tactics, you'll unveil the desired outcome for the variable you're solving for.

Mastering Variable Isolation

Variable isolation is the process of rearranging an equation to get a singular variable on one side. This allows you to determine the value or expression for that variable.
In our equation, \((y-2)(a+1)=x\), isolate \(y\) by systematically performing inverse operations. Begin by removing the multiplication factor, \((a+1)\), by dividing both sides by it, creating a simpler equation: \(y-2 = \frac{x}{a+1}\).
Finally, tackle any constants affecting \(y\). Here, adding 2 isolates \(y\), giving:

• \[ y = \frac{x}{a+1} + 2 \]

This systematic isolation empowers you to derive precise answers and understand how each variable in a problem contributes to the outcome. Master variable isolation, and algebraic solutions become an accessible step-by-step process!