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Chapter 3: Problem 1

Solve the equation and check your solution. $$2(y-4)=0$$

Short Answer

Expert verified
The solution of the equation is \(y = 4\).

Step by step solution

01

Distribute

Firstly, distribute the 2 across the expression within the parentheses: \(2\times y - 2\times4\). This becomes \(2y - 8 = 0\). Now, the equation is simpler.
02

Isolate the y term

To isolate the variable y, add 8 to both sides of the equation: \(2y = 8\).
03

Solve for y

Divide both sides by 2, to finally find the solution for y: \(y = 8 \div 2\) which simplifies to \(y = 4\).
04

Verification

Finally, substitute y=4 into the original equation to check the solution: \(2(4-4) = 0\). Simplifying gives \(0 = 0\), which shows that the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental concept in algebra used to simplify expressions or equations. It helps in distributing a single number over terms within parentheses. In our exercise, we have the equation \(2(y-4)=0\).

The distributive property allows us to multiply the 2 by each term inside the parentheses. So, \(2 imes (y-4)\) becomes \(2 imes y - 2 imes 4\) which simplifies to \(2y - 8\). This turns the equation into \(2y - 8 = 0\).

This property is particularly useful because it turns the equation from a potentially complex form into a simpler linear equation by removing the parentheses.
Isolating Variables
Isolating variables is a key step in solving linear equations. The goal is to isolate the unknown variable on one side of the equation to easily solve for its value. In our case, we have the equation \(2y - 8 = 0\).

To begin isolating \(y\), we perform operations that will cancel out other terms surrounding \(y\) while keeping the equation balanced. Start by adding 8 to both sides, leading to \(2y = 8\).

Next, divide both sides by the coefficient of \(y\), which is 2, resulting in \(y = 4\).

This process ensures that we transform the equation into a form where the variable stands alone, making it easier to identify its value.
Verification of Solutions
Verifying your solution is like checking your work. It's an important step to ensure that the value found actually satisfies the original equation. Once you have found \(y = 4\), substituting it back into the original equation \(2(y-4) = 0\) is crucial.

Replace \(y\) with 4, giving us \(2(4-4) = 0\).
Simplifying inside the parentheses, we get \(2 \times 0 = 0\).

This confirms that both sides of the equation are equal; thus, the solution \(y = 4\) is indeed correct.

Verifying solutions is a good practice as it helps reinforce understanding of the problem-solving process and ensures any errors are caught.

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