Problem 9 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{3 y+1}=4\)

Step-by-Step Solution

Answer

The solution is \(y = 5\).

Step 1: Isolate the Square Root

The equation given is \( \sqrt{3y+1} = 4 \). Since the square root is already isolated on one side of the equation, we can proceed to the next step.

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation: \((\sqrt{3y+1})^2 = 4^2\). This simplifies to \(3y + 1 = 16\).

Step 3: Solve for y

Now, solve the linear equation. Subtract 1 from both sides: \(3y = 15\). Then, divide both sides by 3: \(y = 5\).

Step 4: Verify the Solution

Substitute \(y = 5\) back into the original equation to check if it is a valid solution. \(\sqrt{3(5) + 1} = \sqrt{15 + 1} = \sqrt{16} = 4\). The left-hand side matches the right-hand side, confirming that \(y = 5\) is a correct solution.

Key Concepts

Linear Equations

Linear equations are mathematical statements that involve variables, coefficients, and constants. They usually take the form of ax + b = c, where "x" is the variable. The goal in solving linear equations is to find the value of the variable that makes the equation true.

Why are they so important?

• Linear equations are fundamental in algebra and are the building blocks for understanding more complex mathematics.
• They model real-world situations, helping us to solve problems in science, engineering, and finance.

In our example, after removing the square root in Step 2, we derived a linear equation: 3y + 1 = 16.
We solved this by performing inverse operations, like subtraction and division, to isolate "y".

Square Roots

Square roots are a critical concept in mathematics. When you see a square root symbol (\( \sqrt{} \)), it represents a number which, when multiplied by itself, gives the original number under the root sign.

For example, \( \sqrt{16} = 4 \) because \( 4 \cdot 4 = 16 \). Here's why mastering them is key:

• They help us in mathematical problem solving, especially in quadratic equations and geometry.
• Knowing how to manipulate square roots is crucial for simplifying equations.

In the original exercise, we had \( \sqrt{3y + 1} = 4 \). To solve it, we removed the square root by squaring both sides. This operation is pivotal in turning the problem into a simpler linear equation.

Verification of Solutions

Verification of solutions is a crucial step in problem-solving. After obtaining a potential solution, we must check it against the original equation to ensure its correctness.

Why is this verification step crucial?

• It helps to catch mistakes made during the solving process.
• It ensures that our solution actually satisfies the original condition posed by the equation.

Let’s take the value \( y = 5 \) from the original step-by-step solution. By substituting \( y = 5 \) back into \( \sqrt{3y + 1} \), we calculated \( \sqrt{16} = 4 \), which matches the right side of the equation.

This confirms that our solution of \( y = 5 \) is indeed valid, illustrating the importance of always verifying your answer for accuracy.