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Chapter 3: Problem 27

Solve and check each equation. $$-56=-7 v$$

Short Answer

Expert verified
The solution to the equation \(-56 = -7v\) is \(v=8\). When the obtained value of \(v=8\) is plugged back into the original equation, both sides are equal, confirming that the solution is correct.

Step by step solution

01

Identify the equation

The given equation is \(-56 = -7v\), where 'v' is the variable we need to solve for.
02

Isolate the variable 'v'

To isolate 'v', we must get rid of the coefficient, which is -7 in this case. As -7 is multiplied with 'v', we will divide both sides of the equation by -7 to isolate 'v': \(\frac{-56}{-7} = \frac{-7v}{-7}\)
03

Simplify and find the value of 'v'

Now, simplifying the equation from Step 2, we get: \(8 = v\) Therefore, the value of 'v' is 8.
04

Check the solution

To check our solution, we will plug the obtained value of 'v' (which is 8) back into the original equation and see if both sides are equal: \(-56 = -7(8)\) \(-56 = -56\) Since both sides of the equation are equal, our solution for 'v' is correct and confirms that \(v = 8\).

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Key Concepts

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

Solving Linear Equations
Let's break down the process of solving linear equations, as shown in our example equation: \(-56 = -7v\). A linear equation is simply an equation where the highest power of the variable is one. Solving these equations involves finding the value of the unknown variable.
In this specific case, the unknown variable is \(v\). The key to solving linear equations is to perform operations that simplify the equation, leading to the identification of the variable's value. Here are some steps to guide you:
  • First, identify the equation you are working with, ensuring all terms on one side either involve the variable or are constants.
  • Next, simplify the equation by eliminating constants that don't contain the variable.
  • Finally, perform any necessary operations to isolate the variable completely.
By diligently following these steps, you'll be able to solve any linear equation confidently.
Isolating Variables
Isolating the variable is a pivotal step in solving equations. In our example, the equation was \(-56 = -7v\). The goal is to rearrange the equation to make \(v\) the subject. This means getting \(v\) on one side by itself:
  • Here, \(-7\) is the coefficient of \(v\), meaning it is multiplied by \(v\). To isolate \(v\), perform the opposite operation, which is division in this case.
  • Divide both sides of the equation by \(-7\): \(\frac{-56}{-7} = \frac{-7v}{-7}\)
  • This simplifies to \(v = 8\).
This method of isolating the variable through opposite operations is core to solving many algebraic equations. It ensures the variable stands alone, allowing for direct calculation of its value. Remember, whatever operation you apply to one side of the equation, you must apply to the other side to maintain equality.
Checking Solutions
Checking your solution is the final, crucial step in solving equations. Once you find a value for the variable, substitute it back into the original equation to see if it holds true. Let's apply this to our example:
  • After isolating \(v\), we found \(v = 8\).
  • Substitute \(8\) back into the original equation: \(-56 = -7(8)\).
  • Calculate \(-7(8)\) to verify if both sides equal \(-56\).
  • Indeed, \(-56 = -56\), confirming our solution is correct.
This step is essential because it validates that our operations were accurate and we found the correct value for the variable. It enhances your confidence in handling more complex algebraic problems.

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