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Chapter 11: Problem 35

Solve using the multiplication principle. Don't forget to check! $$ -3.1 y=21.7 $$

Short Answer

Expert verified
The solution is y = -7.

Step by step solution

01

- Isolate the variable

To solve for the variable, we need to isolate y. Since y is currently being multiplied by -3.1, we will divide both sides of the equation by -3.1 to isolate y. Divide both sides by -3.1: calculate: $$ y = \frac{21.7}{-3.1} $$ This simplifies to: $$ y = -7 $$
02

- Check the solution

To verify if the obtained value of y is correct, substitute y back into the original equation and check if the left-hand side equals the right-hand side. Start with the original equation: $$ -3.1y = 21.7 $$ Substitute y = -7: $$ -3.1(-7) = 21.7 $$ Calculate: $$ 21.7 = 21.7 $$ As both sides of the equation are equal, the solution is verified to be correct.

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Key Concepts

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

Solving Linear Equations
Linear equations are equations where the variable (usually denoted as \(x\) or \(y\)) is not raised to any power other than one. These equations form a straight line when graphed.
Solving linear equations involves finding the value of the variable that makes the equation true.
For example, consider the equation \(-3.1 y = 21.7\). This is a linear equation because the variable \(y\) is not raised to any power other than one. The main goal here is to find the value of \(y\) that satisfies this equation.
Here's a quick guide on how to solve linear equations:
  • Identify the variable you need to solve for.
  • Perform the same operation on both sides of the equation to keep it balanced.
  • Isolate the variable on one side of the equation.
  • Simplify the equation.
Isolating Variables
Isolating a variable means getting the variable alone on one side of the equation.
This is crucial because it allows us to solve for the variable.
In the equation \(-3.1 y = 21.7\):
  • The variable \(y\) is being multiplied by \(-3.1\).
  • To isolate \(y\), we need to undo this multiplication by performing the opposite operation, which is division.
  • By dividing both sides of the equation by \(-3.1\), we isolate \(y\).
So, we divide both sides by \(-3.1\):
\( y = \frac{21.7}{-3.1} \)
This simplifies to:
\( y = -7 \)
Now we've successfully isolated \(y\) and solved for it.
Verification of Solutions
Verification of solutions is a critical step in solving equations, ensuring that the obtained solution is correct.
To verify a solution, we need to substitute the solution back into the original equation and see if it satisfies the equation.
Let's verify our solution for \( y = -7 \) in the original equation \( -3.1y = 21.7 \):
  • Start with the original equation: \( -3.1y = 21.7 \)
  • Substitute \( y = -7 \) into the equation: \( -3.1(-7) = 21.7 \)
  • Calculate the left-hand side: \( 21.7 = 21.7 \)
Since both sides of the equation are equal, the solution \( y = -7 \) is verified to be correct.
This step is essential to confirm the accuracy of your solution and ensure that no mistakes were made during the calculation process.

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