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Chapter 3: Problem 32

Solve each equation. Check your solution. $$7 y+3=-11$$

Short Answer

Expert verified
The solution to the equation is \(y = -2\).

Step by step solution

01

Isolate the Variable

We start with the equation \(7y + 3 = -11\). The first step is to isolate the term with the variable by subtracting 3 from both sides of the equation. This gives us: \[7y + 3 - 3 = -11 - 3\] Simplifying this results in: \[7y = -14\]
02

Solve for the Variable

Now that the equation is \(7y = -14\), we can solve for \(y\) by dividing both sides by 7. This gives us: \[y = \frac{-14}{7}\] Simplifying this fraction, we find: \[y = -2\]
03

Check the Solution

To ensure our solution for \(y\) is correct, we substitute \(y = -2\) back into the original equation \(7y + 3 = -11\). Substituting gives: \[7(-2) + 3 = -11\] Simplify the left side: \[-14 + 3 = -11\] Which results in \(-11 = -11\), confirming that our solution is correct.

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Key Concepts

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

Variable Isolation
When solving any linear equation, the first goal is to isolate the variable. This means getting the variable by itself on one side of the equation. For our example equation, \(7y + 3 = -11\), the term with the variable is \(7y\). To isolate this term, we need to eliminate any other numbers on the same side by performing inverse operations. Here, \(+3\) is added to \(7y\), so we subtract 3 from both sides to remove it. This step maintains the balance of the equation, much like a set of scales. Once simplified, the equation becomes \(7y = -14\).
  • Identify the term with the variable
  • Use inverse operations to remove other terms
  • Simplify each side as needed
By following these steps, you've successfully isolated the variable terms from the constants.
Equation Solving Steps
After isolating the variable, the next task is to solve the equation for the variable itself. In our example, the equation \(7y = -14\) has been simplified to a form where we need to find the value of \(y\) that satisfies it.To do this, we need to divide both sides by 7, which is the coefficient of \(y\). This is done to "undo" the multiplication by 7. By performing this division, you scale the equation such that \(y\) stands alone: \[y = \frac{-14}{7}\] Simplify the fraction to find:\(y = -2\)
  • Check the simplified equation to identify the operation needed to isolate \(y\)
  • Perform the inverse operation
  • Simplify to find the value of the variable
These steps will help you find the value that balances the equation, solving for the unknown.
Checking Solutions
Once we have a solution, it's always smart to check it. This step reduces mistakes and increases confidence in your answer. Let's substitute \(y = -2\) back into the original equation \(7y + 3 = -11\) to verify.Substitute:\[7(-2) + 3 = -11\]Simplify the left side:\[-14 + 3 = -11\]And you'll find that both sides equal \(-11\), proving that \(y = -2\) is indeed the correct solution.
  • Substitute the solution back into the original equation
  • Simplify each side to see if they match
  • Confirm equality to verify the solution
By following these practices, you ensure that your solution is accurate, giving you a solid verification of your findings.

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