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Chapter 3: Problem 46

Solve and check each equation. $$2 y-5=3$$

Short Answer

Expert verified
The short answer to the equation \(2y - 5 = 3\) is \(y = 4\).

Step by step solution

01

Add 5 to both sides of the equation

We want to get the variable y isolated on one side of the equation. To do this, we need to get rid of the -5 on the left side. We can add +5 to both sides to counteract the -5. \(2y - 5 + 5 = 3 + 5\) This simplifies to: \(2y = 8\)
02

Divide by 2

Now that we have \(2y = 8\), we can divide both sides by 2 to get y by itself. \( \frac{2y}{2} = \frac{8}{2} \) This simplifies to: \(y = 4\)
03

Check the solution

To check our solution, we can subsitute the value of y back into the original equation: \(2(4) - 5 = 3\) This simplifies to: \(8 - 5 = 3\) \(3 = 3\) Since both sides of the equation are equal, our solution \(y = 4\) is correct.

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Key Concepts

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

Addition Property of Equality
Solving linear equations often requires rearranging terms to isolate the variable. One such technique is the addition property of equality.
This principle states that adding the same number to both sides of an equation preserves equality.
In our exercise, we faced the equation \(2y - 5 = 3\).
The goal was to isolate \(y\). The subtraction \(-5\) on the left side needed to be eliminated.
By adding \(+5\) to both sides, we counteracted the \(-5\) effectively balancing the equation:
  • Original equation: \(2y - 5 = 3\)

  • Add \(+5\) to both sides: \(2y - 5 + 5 = 3 + 5\)

  • Simplified result: \(2y = 8\)
This method ensures the integrity of the equation remains intact, bringing us closer to finding the variable's value.
Division Property of Equality
Once we've used the addition property to simplify the equation, the next step is often to use the division property of equality.
This property states that dividing both sides of an equation by the same non-zero number keeps the equation balanced.
When we have an equation like \(2y = 8\), the aim is to further isolate \(y\).
This can be done by dividing each side of the equation by 2:
  • Equation before division: \(2y = 8\)

  • Divide both sides by 2: \(\frac{2y}{2} = \frac{8}{2}\)

  • Resulting equation: \(y = 4\)
Using the division property of equality helped reduce the coefficient of \(y\) to 1, allowing us to identify \(y\)'s precise value.
Checking Solutions in Algebra
It's crucial to verify that your solution is correct in algebra, ensuring accuracy and understanding.
To check our solution for the equation \(2y - 5 = 3\), we substitute our calculated \(y = 4\) back into the original equation.
Substitution is an excellent method to confirm correctness:
  • Calculate \(2(4) - 5\)

  • Simplify: \(8 - 5 = 3\)

  • Compare: \(3 = 3\)
Both sides of the equation match, affirming that our solution \(y = 4\) is indeed correct.
This step completes the problem-solving process by ensuring confidence in our answer and in our understanding of the equation's mechanics.

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