Problem 37 - Algebra for College Students

Question

For Problems 37-46, solve each equation for the indicated variable. \(2 x-5 y=7\) for \(x\)

Step-by-Step Solution

Answer

The solution for \(x\) is \(x = \frac{5y + 7}{2}\).

Step 1: Isolate the Term Containing the Variable

Start with the equation \(2x - 5y = 7\). To solve for \(x\), first add \(5y\) to both sides of the equation to isolate the term with \(x\): \[2x = 5y + 7\].

Step 2: Solve for the Variable

Now, divide both sides of the equation by 2 in order to solve for \(x\): \[x = \frac{5y + 7}{2}\].

Key Concepts

Linear Equations

Linear equations are equations where variables appear only to the first power, without any exponents. These equations form a straight line when graphed. They typically take the general form:

• For one variable: \(ax + b = c\)
• For two variables: \(ax + by = c\)

Each variable is multiplied by a constant, and these constants help define the slope and position of the line.
The beauty of linear equations lies in their simplicity, making them a foundational concept in algebra. They are used to solve practical problems involving relationships between quantities. Understanding how to manipulate and solve these equations is key to mastering more complex algebraic problems.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to rearrange equations. The goal is to simplify an expression or solve an equation.
Common techniques include:

• Adding or subtracting the same term from both sides
• Multiplying or dividing both sides by the same non-zero number

These operations keep the equation balanced, much like a seesaw. By strategically choosing which operations to apply, you can isolate variables and solve equations more easily.
For example, in the equation \(2x - 5y = 7\), adding \(5y\) to both sides is an effective move to bring the equation closer to a form where \(x\) can be easily isolated. Remember, the key is to perform operations equally on both sides of the equation to maintain equality.

Isolation of Variables

The isolation of variables is a method used to solve an equation for a particular variable. This often involves multiple steps, as seen in the original exercise.
The process usually follows these patterns:

• Move all terms containing the variable of interest to one side of the equation
• Other terms should be shifted to the opposite side

Once the target variable is isolated, you can find its value by performing any necessary operations.
In the example \(2x - 5y = 7\) when solving for \(x\), we first move \(5y\) to the other side to leave \(2x\) alone. Then, we simply divide by 2 to isolate \(x\) completely: \( x = \frac{5y + 7}{2}\)
Mastering isolation techniques is vital for systematically solving equations and understanding algebraic expressions.