Problem 55 - Elementary Algebra

Question

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ 3 x+7 y=9 \quad \text { for } x $$

Step-by-Step Solution

Answer

\(x = \frac{9 - 7y}{3}\).

Step 1: Isolate Terms Involving x

Start by moving all the terms that don't involve the variable \(x\) to the other side of the equation. You have the equation \(3x + 7y = 9\). Subtract \(7y\) from both sides to isolate the \(3x\) term: \[ 3x = 9 - 7y \].

Step 2: Solve for x

Now that the \(x\) terms are isolated, solve for \(x\) by dividing all terms in the equation by the coefficient of \(x\), which is 3: \[x = \frac{9 - 7y}{3}\].

Key Concepts

Isolate Variable

When solving linear equations, one of the primary goals is to isolate the variable you are solving for. In our example, we are tasked with solving for the variable \(x\). To isolate a variable means to have that variable alone on one side of the equation, making it easy to "read" its value from the equation. This process is often the first crucial step in problem-solving.
The isolation process involves moving all other terms to the opposite side of the equation. In mathematical terms, this means using inverse operations to "undo" what has been done to the variable. For instance, if a number is added to the variable, you must subtract it from both sides of the equation, maintaining the balance.
In the given exercise, the term \(7y\) is added to \(3x\). To isolate \(x\), we subtract \(7y\) from both sides of the equation, transforming it into \(3x = 9 - 7y\). This step ensures that \(x\) is grouped on one side, ready for further manipulation.

Linear Equations

Linear equations form the backbone of algebra. These equations represent straight lines when plotted on a graph and take the standard form \(ax + by = c\). In these equations, both variables are raised to the power of one, meaning they appear in their simplest form and do not involve squares or any other exponents.
The simplicity of linear equations is in their structure. They are predictable and uniform, which makes isolating variables straightforward. Solving these equations usually involves the basic operations: addition, subtraction, multiplication, and division.
Our exercise presents a classic linear equation \(3x + 7y = 9\). Such equations often require solving for a specific variable, which is beautifully manageable due to their linear nature. Solutions provide a consistent set of rules and can easily translate into real-world problem-solving scenarios.

Algebraic Manipulation

In solving equations, algebraic manipulation is the key process that alters the original form of the equation to attain a solution. This involves using algebraic rules and operations to both simplify and solve equations. Techniques include distributing terms, combining like terms, and applying the distributive, associative, and commutative properties.
To continue from our isolated variable \(3x = 9 - 7y\), algebraic manipulation allows us to solve for \(x\). By dividing every term by 3, which is the coefficient of \(x\), we simplify the equation to \(x = \frac{9 - 7y}{3}\). This form provides a neat expression for \(x\), fully isolated and ready to be evaluated if \(y\) takes on specific values.
Algebraic manipulation also serves as a versatile skill in mathematics, allowing transformations and solutions to emerge from seemingly complex problems. Mastery of this technique empowers students to approach a variety of mathematical challenges with confidence.