Problem 28 - Intermediate Algebra

Question

Solve. \(\sqrt{3 x+1}-2=0\)

Step-by-Step Solution

Answer

x = 1

Step 1: Isolate the Square Root Term

Start by adding 2 to both sides of the equation to isolate the square root term. This gives: \[ \sqrt{3x + 1} = 2 \]

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation. This results in: \[ (\sqrt{3x + 1})^2 = 2^2 \]So the equation becomes: \[ 3x + 1 = 4 \]

Step 3: Solve the Linear Equation

Subtract 1 from both sides to start solving for x:\[ 3x = 3 \]Next, divide both sides by 3:\[ x = 1 \]

Key Concepts

Linear Equations

At the heart of many algebra problems, we find the linear equation. A linear equation is simply an equation that represents a straight line when graphed. It takes the general form of \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants. Linear equations are called 'linear' because their graph is a straight line, and they have a constant rate of change.
These equations are significant in algebra largely because they are straightforward to solve using basic arithmetic operations. Solving them involves a series of steps to isolate the variable. In practice, this often involves adding, subtracting, multiplying, or dividing both sides of the equation by the same number to get the variable alone on one side.

• Example: For \(3x + 1 = 4\), subtract 1 from both sides to move the +1 to the other side, getting \(3x = 3\).
• Then, divide by 3 to solve for \(x\), resulting in \(x = 1\). This step-by-step breakdown shows how versatile and manageable linear equations can be.

Understanding linear equations is foundational for grasping more complex algebraic concepts and is a key skill in problem-solving across various applications.

Isolating the Variable

Isolating the variable is a crucial first step in solving equations, particularly when you're faced with more complicated expressions like those involving square roots. The goal is to manipulate the equation so that the variable you're solving for is alone on one side of the equal sign. This makes it easier to perform further operations that lead to a solution.
In the context of our exercise, isolating the square root term involves rearranging the given equation, \(\sqrt{3x + 1} - 2 = 0\). Here, you add 2 to both sides, resulting in \(\sqrt{3x + 1} = 2\).

• The idea is to change the equation's form without altering its truth, maintaining balance by performing the same operation on both sides.
• Once the term is isolated, you can proceed to eliminate the square root by squaring both sides, simplifying the equation further.

This step is pivotal because it sets the stage for applying other mathematical operations to solve the equation. By isolating the variable, we make subsequent actions more straightforward and logical.

Squaring Both Sides

Squaring both sides of an equation is a powerful technique used when dealing with square roots. The objective is to remove the square root, transforming the equation into a linear form, which is easier to solve. After isolating the square root term, you are left with an equation like \(\sqrt{3x + 1} = 2\).
To eliminate the square root, you must square both sides of the equation:
\[(\sqrt{3x + 1})^2 = 2^2\]

• This operation converts \(\sqrt{3x + 1}\) into \(3x + 1\), and \(2^2\) results in 4.
• The equation now reads \(3x + 1 = 4\), a simple linear equation that can easily be solved.

This concept is crucial, as it turns the equation into a familiar form that easily leads to a solution. It’s important to handle the squaring process carefully, especially if there are other terms or operations present in the original equation.