Problem 19 - Elementary Algebra

Question

For the following problems, solve the equations. $$ \sqrt{3 a+1}=4 $$

Step-by-Step Solution

Answer

Question: Solve for 'a' in the equation $$\sqrt{3a+1} = 4$$. Answer: The solution to the equation is $$a = 5$$.

Step 1: Square both sides of the equation

We'll start by squaring both sides of the equation to eliminate the square root. $$ (\sqrt{3a+1})^2 = (4)^2 $$

Step 2: Simplify

Now, let's simplify the equation. $$ 3a + 1 = 16 $$

Step 3: Isolate the variable 'a'

Next, we want to isolate 'a'. To do this, we first subtract 1 from both sides of the equation. $$ 3a = 15 $$

Step 4: Solve for 'a'

Finally, we'll divide both sides by 3 to find the value of 'a'. $$ a = \frac{15}{3} $$ $$ a = 5 $$ The solution to the equation is $$a = 5$$.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that requires you to find the value of the unknown variable that makes the equation true. In the given problem, the original equation is \(\sqrt{3a+1} = 4\). Understanding how to solve such an equation involves a series of logical steps.

When solving equations, particularly those involving roots, it's important to remember:

• The operation you use must be applied to both sides of the equation to maintain balance.
• Start by removing operations closest to the variable; for roots, this typically means squaring both sides.
• After simplification, further isolate the variable by undoing additional operations like addition or multiplication.

These steps guide you through converting complex expressions into simpler ones until the solution is apparent.

Square Roots

Square roots are a type of radical expression and represent the value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because \(4 \times 4 = 16\).

When faced with an equation involving square roots, like \(\sqrt{3a+1} = 4\), your goal is often to eliminate the square root by squaring both sides. Squaring is the inverse operation of taking a square root.

Upon squaring both sides:

• \((\sqrt{3a+1})^2 = 4^2\) simplifies to \(3a+1 = 16\).
• This transformation removes the square root and lays the foundation for further simplification.

Carefully transforming these root expressions enables you to work with linear forms of the unknown, making it easier to solve.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to rearrange an equation in a form that readily reveals the unknown variable. This is extremely useful when solving equations.

After squaring the original equation in the example problem, you simplify it to \(3a + 1 = 16\). The next steps include:

• Subtract \(1\) from both sides to isolate terms involving \(a\), resulting in \(3a = 15\).
• Divide both sides by \(3\) to solve for \(a\), yielding \(a = 5\).

By performing these operations systematically, you extract the solution while ensuring the equation remains balanced. Each manipulation uses basic arithmetic to simplify the equation step-by-step until the variable is isolated.