Problem 33 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the given equation. $$ \sqrt{3 x+1}=2 $$

Step-by-Step Solution

Answer

The solution to the equation \(\sqrt{3x + 1} = 2\) is \(x = 1\).

Step 1: Square both sides of the equation

To eliminate the square root, square both sides of the equation. This will give us: \[ (\sqrt{3x + 1})^2 = (2)^2 \]

Step 2: Simplify the equation

When we square the square root, it cancels out, leaving us with: \[ 3x + 1 = 4 \]

Step 3: Isolate the variable x

Subtract 1 from both sides and then divide by 3: \[ 3x = 3 \Rightarrow x = 1 \] The solution to the equation \(\sqrt{3x + 1} = 2\) is \(x = 1\).

Key Concepts

Solving Equations

To solve equations, we look for the value of the variable that makes the equation true. An equation is like a balance scale, so any action on one side of the equation must be done to the other side. This keeps everything balanced. For our example, \(\sqrt{3x + 1} = 2\), the goal is to find \(x\) such that when substituted back into the equation, both sides are equal. We do this in steps:

• Identifying the Equation: The equation here involves a square root, making it slightly different from simpler linear equations.
• Manipulation: Typically, squaring both sides can help remove the square root, isolating the terms with \(x\).

In essence, solving equations requires a clear approach to isolate the variable through various algebraic methods, keeping the equation balanced.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. In algebraic terms, if \(y^2 = x\), then \(y\) is a square root of \(x\). Square roots appear often in equations, and dealing with them typically involves:

• Understanding the Radical Notation: The symbol \(\sqrt{}\) represents the square root.
• Removing Square Roots: This usually involves squaring both sides of an equation to eliminate the square root, as seen in \((\sqrt{3x + 1})^2 = 4\).

This remains a fundamental tool in simplifying equations to allow for easier manipulation and solution. It’s important to note that squaring eliminates potential negative results, which makes checking your solution crucial in some contexts.

Linear Equations

A linear equation is straightforward, typically in the form of \(ax + b = c\). They are called linear because when graphed, they make a straight line. For our problem, here's how we approached it:

• Simplification: After squaring both sides, the equation \(3x + 1 = 4\) presented a linear form.
• Solving for \(x\): We subtracted 1 from both sides, leading us to \(3x = 3\), then divided by 3 to find \(x = 1\).

Linear equations involve basic arithmetic operations and are the foundation of solving more complex algebraic equations. Once in this form, solving them is systematic and direct. This typically involves isolating the variable by reversing operations.