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Chapter 2: Problem 37

Solve and check. Label any contradictions or identities. $$2-5 y=26-y$$

Short Answer

Expert verified
The solution is \(y = -6\) and it is correct because \(32 = 32\) when substituted back into the original equation.

Step by step solution

01

Move Variables to One Side

Add \(y\) to both sides of the equation to move all terms with \(y\) from the right side of the equation to the left side. \(2 - 5y + y = 26 - y + y\)
02

Simplify Both Sides

Combine like terms on both sides. \(2 - 4y = 26\)
03

Isolate the Variable

Subtract 2 from both sides to isolate the term with \(y\). \(2 - 2 - 4y = 26 - 2\) which simplifies to \(-4y = 24\)
04

Solve for \(y\)

Divide both sides by -4 to solve for y. \(\frac{-4y}{-4} = \frac{24}{-4}\) \(y = -6\)
05

Check the Solution

Substitute the value of \(y\) back into the original equation to verify the solution. \(2 - 5(-6) = 26 - (-6)\) This simplifies to \(2 + 30 = 26 + 6\) \(32 = 32\), which is true.

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Key Concepts

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

Isolating Variables
Isolating variables means getting the variable you're solving for by itself on one side of the equation. This makes it easier to find the value of that variable. In our exercise, the variable is \(y\).

To start, we need to move all the terms with \(y\) to one side of the equation. We began with the equation \(2-5y=26-y\). The first step was adding \(y\) to both sides to get all \(y\) terms on the left side:

$$ 2 - 5y + y = 26 - y + y $$

This step is crucial. By placing all \(y\) terms on one side, we simplify further steps. Isolating the variable makes it straightforward to solve for \(y\), giving us a clear path to follow.
Simplifying Equations
Simplifying equations involves combining like terms and eliminating constants as much as possible. This helps to make the equation easier to solve. After moving the terms around, our equation changed to:

$$ 2 - 4y = 26 $$

Here, we鈥檝e combined the \(-5y\) and \(y\) terms. The next step is to isolate the term with \(y\). We subtract 2 from both sides of the equation:

$$ 2 - 2 - 4y = 26 - 2 $$

Simplifying further, we obtain:

$$ -4y = 24 $$

In this form, our equation is simpler and only contains the \(y\) term. Simplifying equations step-by-step makes it easier to isolate the variables and solve for them accurately.
Checking Solutions
To ensure our solution is correct, we need to check it by substituting the variable back into the original equation. Checking solutions helps to verify if the value found satisfies the equation. For \(y = -6\), substituting it back into the original equation:

$$ 2 - 5(-6) = 26 - (-6) $$

This simplifies as follows:

$$ 2 + 30 = 26 + 6 $$

Giving us:

$$ 32 = 32 $$

Since both sides of the equation are equal, this confirms our solution. Checking solutions prevents errors and ensures the value found is indeed correct. Verifying the solution reinforces understanding and ensures accuracy in solving equations.

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