Problem 62 - Elementary Algebra

Question

Solve by extracting the roots. $$ 2(3 y-13) 2-52=0 $$

Step-by-Step Solution

Answer

The solution is \( y = 13 \).

Step 1: Expand the Equation

First, let's expand the equation given. We have: \[ 2(3y - 13) - 52 = 0 \] Multiply 2 by each term inside the parentheses:\[ 2 \cdot 3y - 2 \cdot 13 = 6y - 26 \] Replace the left side of the equation with this expression:\[ 6y - 26 - 52 = 0 \]

Step 2: Simplify the Equation

Now, simplify the left side of the equation. Combine like terms: \[ 6y - 26 - 52 = 6y - 78 \] So the equation simplifies to: \[ 6y - 78 = 0 \]

Step 3: Isolate the Variable

To solve for \( y \), add 78 to both sides of the equation: \[ 6y - 78 + 78 = 78 \] This reduces to: \[ 6y = 78 \]

Step 4: Solve for \( y \)

Finally, divide both sides of the equation by 6 to isolate \( y \): \[ y = \frac{78}{6} \] Calculate the division: \[ y = 13 \]

Key Concepts

Solving Equations

When solving algebraic equations, the primary goal is to find the value of the unknown variable that makes the equation true. This process involves systematically manipulating the equation to simplify and eventually isolate the variable. In our example, the equation is initially given as \(2(3y - 13) - 52 = 0\).

Here's how we solve it:

• Expand or distribute any expressions to simplify the equation where needed. For example, we distribute the 2 across \((3y - 13)\) to simplify to \(6y - 26\).
• Combine like terms to further simplify the equation. After distribution, subtracting 52 combines to get \(6y - 78 = 0\).
• Continue simplifying and solving using inverse operations to unravel any additional steps needed. In our case, adding 78 to both sides leaves us with \(6y = 78\).

The final steps will always aim to isolate the unknown variable for easy solving.

Isolation of Variables

Isolation of variables is a crucial step in solving equations. This means altering the equation so that the variable is by itself on one side of the equation. In our exercise, we want to isolate \(y\) in \(6y = 78\).

To achieve isolation:

• Identify and apply inverse operations, such as addition, subtraction, multiplication, or division. Each operation will help "undo" the equation's current state.
• Add 78 to both sides when we initially have \(6y - 78 = 0\), which helps move numbers away from the variable.
• Finally, divide both sides by 6, which originally multiplied \(y\), giving us \(y = 13\).

By isolating \(y\), we find its value straightforwardly, which in this example is 13.

Simplification of Expressions

Simplification of expressions involves combining like terms and making an equation more manageable, without changing its value. It's a vital skill for handling complex algebraic equations.

In the provided problem, simplification is achieved as follows:

• Expand the expressions using distributive property and combine similar (like) terms together.
• After expanding, the expression in question initially becomes \(6y - 26 - 52\).
• Combine the constants: \(-26\) and \(-52\), yielding \(6y - 78\).

This streamlined form helps in not only making calculations easier but also clearer and more direct when it comes to isolating the variable. Simplifying properly is key to understanding what steps to take next in solving equations.