Problem 45 - Algebra for College Students

Question

Solve each equation for the indicated variable. \((y+1)(a-3)=x-2\) for \(y\)

Step-by-Step Solution

Answer

\(y = \frac{x + 1 - a}{a - 3}\)

Step 1: Expand the equation

To solve for \(y\), we need to first expand the left side of the equation. Distribute \((y+1)\) across \((a-3)\): \((y+1)(a-3) = ya - 3y + a - 3\). The equation now becomes: \(ya - 3y + a - 3 = x - 2\).

Step 2: Isolate terms involving y

To solve for \(y\), collect all terms containing \(y\) on one side of the equation. Move the other terms to the right side: \(ya - 3y = x - 2 - a + 3\). Simplify the right side: \(ya - 3y = x + 1 - a\).

Step 3: Factor out y

Factor \(y\) from the left side: \(y(a - 3) = x + 1 - a\). This step prepares the equation for solving \(y\) by making \(y\) a single term.

Step 4: Solve for y

Divide both sides of the equation by \((a - 3)\) to solve for \(y\): \(y = \frac{x + 1 - a}{a - 3}\). This isolates \(y\) and is the final solution.

Key Concepts

Expanding Equations

When it comes to solving equations, one of the initial steps is often expanding them. This involves taking the products of terms enclosed in parentheses and distributing them. It makes the equation easier to handle by simplifying it into a more manageable form.

In our exercise, we started with (y+1)(a-3)=x-2. To expand this, apply the distributive property: multiply (y+1) by each part of (a-3), so that you get ya - 3y + a - 3.

Think of expanding as opening up a gift; it allows us to see the components more clearly, setting the stage for the next steps.

Isolating Variables

Once the equation is expanded, the next task is to isolate the variable we are solving for. In our case, it's y. Imagine isolating variables as tidying a room; you gather similar items together to make things organized.

For terms involving y, the goal is to get them on one side. From the expanded equation, we organize by moving non- y terms to the opposite side: ya - 3y = x + 1 - a. Now, everything with y is neatly on one side, preparing for the next step, which is factoring.

Factoring

Factoring is like simplifying a complex maze into an easy path. In solving equations, it helps to consolidate terms and reduce complexity.

In our exercise, once the y terms are collated, factor out y from ya and -3y. This gives us y(a - 3). The rest of the expression remains unchanged as x + 1 - a.

Factoring effectively weeds out unnecessary complexity, ensuring our focus remains solely on the variable in question and preparing us for a straightforward solution.

Step-by-Step Solutions

Breaking down a math problem into clear, manageable steps is crucial for understanding. Problem-solving becomes much less daunting and more logical.

Here’s a recap of the steps involved in the original exercise:

• Expand to simplify the equation.
• Isolate the variable, gathering all y terms together.
• Factor to clear all non-essential variables affecting y.
• Solve by making y the subject directly.

Each step builds on the previous one, leading to a comprehensive and clear solution: y = \(\frac {x + 1 - a}{a - 3}\).

Taking a methodical approach ensures every aspect of the problem is addressed systematically, leading to success in most mathematical endeavors.