Problem 415 - Elementary Algebra

Question

In the following exercises, solve. $$ 3 \sqrt{7 y+1}-10=8 $$

Step-by-Step Solution

Answer

y = 5

Step 1 - Isolate the Square Root

First, add 10 to both sides of the equation to isolate the square root term. \[ 3 \sqrt{7y + 1} - 10 + 10 = 8 + 10 \] Simplifying this, we get: \[ 3 \sqrt{7y + 1} = 18 \]

Step 2 - Eliminate the Coefficient of the Square Root

Divide both sides by 3 to get rid of the coefficient of the square root. \[ \frac{3 \sqrt{7y + 1}}{3} = \frac{18}{3} \] Simplifying this, we get: \[ \sqrt{7y + 1} = 6 \]

Step 3 - Square Both Sides

To eliminate the square root, square both sides of the equation. \[ (\sqrt{7y + 1})^2 = 6^2 \] This gives us: \[ 7y + 1 = 36 \]

Step 4 - Solve for y

Subtract 1 from both sides to isolate the term with y. \[ 7y + 1 - 1 = 36 - 1 \] Simplifying: \[ 7y = 35 \] Divide both sides by 7 to solve for y: \[ \frac{7y}{7} = \frac{35}{7} \] So, \[ y = 5 \]

Key Concepts

Isolating Variables

When solving equations, a key step is isolating the variable. This often means getting the variable term by itself on one side of the equation. In our example: \[ 3 \, \sqrt{7y + 1} - 10 = 8 \] To isolate the square root term, we start by adding 10 to both sides: \[ 3 \, \sqrt{7y + 1} = 18 \] Now the term with the variable is alone on one side of the equation. This sets up the equation for further steps. Always remember:

• Perform the same operation on both sides.
• Use inverse operations to 'undo' additions, subtractions, multiplications, or divisions.

These skills are the backbone of algebraic manipulation, making complex problems manageable.

Square Roots

Square roots can seem intimidating, but handling them carefully simplifies your math journey. In the equation: \[ 3 \, \sqrt{7y + 1} = 18 \] we must eliminate the square root to solve for 'y'. First, divide both sides by 3 to simplify: \[ \sqrt{7y + 1} = 6 \] Next, square both sides, because squaring undoes the square root: \[ (\sqrt{7y + 1})^2 = 6^2 \] Simplifying, this equation becomes: \[ 7y + 1 = 36 \] Now, the equation is free of square roots, making it easier to solve. Key points to remember:

• Square roots can be eliminated by squaring.
• Ensure equations are simplified before applying advanced operations.

These tips will help you tackle square roots confidently.

Algebraic Manipulation

Algebraic manipulation involves transforming an equation into a simpler form to find the solution. This often includes combining like terms, using the distributive property, and employing inverse operations. Let's look at the final steps of our example: After isolating the variable and removing the square root, we have: \[ 7y + 1 = 36 \] Subtract 1 from both sides to isolate the term with 'y': \[ 7y = 35 \] Now, divide both sides by 7: \[ y = 5 \] These steps break down the problem into manageable parts. Remember:

• Simplify step-by-step to avoid mistakes.
• Verify each transformation to ensure accuracy.
• Use basic arithmetic to isolate and solve for the variable.

Mastering algebraic manipulation lets you handle more complex equations with ease.