Problem 49 - Precalculus

Question

Solve each equation for \(x\) in terms of the other letters. $$3 a x-2 b=b+3$$

Step-by-Step Solution

Answer

\(x = \frac{b + 1}{a}\)

Step 1: Simplify both sides

Start by simplifying the equation if needed. Here, the equation is already simple: \(3ax - 2b = b + 3\).

Step 2: Isolate terms with x

Move the constant terms to the other side of the equation. Add \(2b\) to both sides to get \(3ax = b + 3 + 2b\), which simplifies to \(3ax = 3b + 3\).

Step 3: Solve for x

Divide each side by \(3a\) to solve for \(x\). This yields \(x = \frac{3b + 3}{3a}\).

Step 4: Simplify the expression

Simplify the fraction \(x = \frac{3(b + 1)}{3a}\) by canceling \(3\) from the numerator and the denominator, giving \(x = \frac{b + 1}{a}\).

Key Concepts

Solve for x

Solving for \(x\) in an algebraic equation means finding the value of \(x\) that satisfies the equation. In the given equation \(3ax - 2b = b + 3\), our goal is to express \(x\) in terms of other variables. To solve for \(x\), follow these steps:

• First, rearrange and simplify the equation so that all terms involving \(x\) are on one side of the equation.
• Next, isolate \(x\) to find its value or expression.

These steps are applied logically to work towards unraveling \(x\). This method helps in understanding the relationship between the variables involved.

Simplifying Expressions

Simplifying an expression is a critical part of solving algebraic equations. This involves combining like terms and performing any basic arithmetic operations needed to make the equation easier to handle. In our initial equation \(3ax - 2b = b + 3\), simplification was not initially necessary until later steps. However, after moving the terms and making an addition like \(3ax = 3b + 3\), you can further simplify it to \(3ax = 3(b + 1)\). Simplification is essential because it provides a clearer path to finding the solution, by creating simpler expressions that are easier to manipulate. Remember, always combine like terms and simplify fractions when possible.

Isolate Variable

The term "isolate the variable" frequently appears in solving algebraic equations. This step involves reorganizing the equation so that the variable of interest stands alone on one side of the equation. In our equation: \[3ax = 3b + 3\] we aimed to isolate \(x\) by dividing both sides by \(3a\), giving \(x = \frac{3b + 3}{3a}\). By isolating the variable, we are essentially simplifying the equation to a form that reveals \(x\)'s relationship to other variables and operations. This step often includes reverse operations such as addition to subtraction, or multiplication to division.

Step-by-Step Solution

Following a step-by-step solution makes solving equations less daunting and more systematic. It involves tackling the problem incrementally, using logical procedures. Here’s how our problem is solved step-by-step:

• Step 1: Simplification, although our starting equation is simple.
• Step 2: Moving constants or terms not involving \(x\) to simplify the equation \(3ax = 3b + 3\).
• Step 3: Dividing both sides by \(3a\) to find \(x = \frac{3b + 3}{3a}\).
• Step 4: Simplifying further if needed, which results in \(x = \frac{b + 1}{a}\).

Using a step-by-step approach helps in minimizing errors and ensures that each part of the equation is dealt with accurately.