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Chapter 2: Problem 70

Solve each equation. Check all solutions. $$\frac{v-7}{3}=-1$$

Short Answer

Expert verified
The solution to the equation \(\frac{v-7}{3} = -1\) is \(v = 4\).

Step by step solution

01

Clear the fraction

To clear the fraction, multiply both sides of the equation by 3. So, the equation transforms into \(3*\frac{v-7}{3} = 3*(-1)\). By doing this, the '3' on the left side of the equation will cancel out with the '3' in the denominator of the fraction, we are left with \(v - 7 = -3\).
02

Isolate variable v

To isolate the variable 'v', add '7' to both sides of the equation. After adding '7', the transformed equation will be \(v - 7 + 7 = -3 + 7\). The '-7' on the left side will cancel out with '+7', making \(v = 4\).
03

Check solution

Insert the found value of 'v' back into the original equation to verify if it's accurate. Substitute 'v = 4' into the original equation \(\frac{v-7}{3} = -1\), then the equation transforms into \(\frac{4 - 7}{3} = -1\). After performing the calculation, the left side of the equation equals '-1', which matches the right side of the equation. Hence, the solution is correct. The value 'v = 4' is the solution for the equation.

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Key Concepts

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

Checking Solutions
After solving an equation, it is crucial to check if your solution is correct by substituting it back into the original equation. This step helps ensure that you have not made any mistakes.
For example, when we find that \( v = 4 \) for the equation \( \frac{v-7}{3} = -1 \), you need to substitute \( 4 \) back into the equation:
  • Replace \( v \) with \( 4 \): \( \frac{4-7}{3} \)
  • Calculate \( 4-7 = -3 \)
  • Then, \( \frac{-3}{3} = -1 \)
If both sides of the equation are equal, the solution is confirmed to be correct. Checking your solution is a good habit to develop, as it helps catch any errors early and strengthens your understanding of the problem.
Fractions in Equations
Fractions in equations can make them seem more complex at first glance. However, they become much simpler once you clear the fractions.
In this exercise, for the equation \( \frac{v-7}{3} = -1 \), the fraction is solved by performing the following steps:
  • Identify the fraction \( \frac{1}{3} \).
  • Multiply both sides of the equation by \( 3 \) to clear the fraction: \( 3*\frac{v-7}{3} = 3*(-1) \)
  • This simplification results in the equation \( v - 7 = -3 \).
Clearing the fraction simplifies the equation and allows you to focus on isolating the variable.
Always know that you can multiply both sides of an equation by the denominator to eliminate the fraction, making the equation easier to solve.
Isolating Variables
The process of isolating a variable involves getting the variable, such as \( v \), on one side of the equation by itself. This is often necessary to solve the equation.
In this case, after clearing the fraction, we have the equation \( v - 7 = -3 \). To isolate \( v \):
  • Add \(7\) to both sides of the equation: \( v - 7 + 7 = -3 + 7 \).
  • On the left side, the \(-7\) and \(+7\) cancel each other out, leaving \( v \).
  • On the right side, calculate \(-3 + 7 = 4\).
This results in \( v = 4 \). Isolating the variable is the key step in solving the equation.
Remember, the goal is to have the variable by itself, which indicates that you have found the solution to the equation.

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