Problem 55 - Precalculus Student Solutions Manual 5th

Question

Solve each equation for \(x\). $$a^{2} x+3 x=2 a^{2}$$

Step-by-Step Solution

Answer

x = \frac{2a^{2}}{a^{2} + 3}

Step 1 Title - Combine Like Terms

First, combine the like terms involving x on the left side of the equation. The initial equation is: \[a^{2} x + 3x = 2a^{2}\] Factor out x from both terms: \[x(a^{2} + 3) = 2a^{2}\]

Step 2 Title - Isolate the Variable

Next, isolate x by dividing both sides of the equation by \(a^{2} + 3\): \[x = \frac{2a^{2}}{a^{2} + 3}\] This gives the value of x in terms of \(a^{2}\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving equations. It involves rearranging and simplifying expressions to make solving for the variable easier. In our exercise, the initial equation is:
\[a^{2} x + 3x = 2a^{2} \]
Our first step is to combine like terms. By identifying the terms that contain the variable x, we can simplify our equation. Both \a^2 x\ and 3x include x, so we factor x out:
\[a^{2} x + 3x = x (a^{2} + 3) \]
This is a key maneuver in algebraic manipulation, simplifying further steps of isolating the variable. With practice, these steps become intuitive, making it easier to solve increasingly complex equations.

Factoring

Factoring is an essential technique that simplifies expressions and equations. Factoring means to express an equation as a product of its factors, which can sometimes make the process of solving the equation easier. In our example, we factor out x from the terms on the left side:
\[a^{2} x + 3x = x(a^{2} + 3) \]
Factoring sets up the equation nicely for the next step. By factoring, we transform the original equation into a simpler format that highlights the structure, making it easier to isolate x. Understanding how and when to factor terms is a powerful tool in algebra that reveals the underlying relationships between the terms. It's particularly useful in solving equations and simplifying expressions.

Isolating Variables

Isolating the variable is the final step in solving an equation, aimed at getting the variable by itself on one side of the equation. In our exercise, after factoring out x, we have:
\[x(a^{2} + 3) = 2a^{2} \]
To isolate x, we divide both sides by the coefficient of x, which is \a^2 + 3\:
\[x = \frac{2a^{2}}{a^{2} + 3} \]
This step reveals the value of x in terms of the other variable (a in this case). Isolating the variable is typically the end goal when solving an equation, because it allows us to express the solution clearly. Through practice, isolating variables becomes second nature, and it's a versatile method that can be applied to various types of equations.