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Chapter 2: Problem 80

Solve each equation. Then check the result. $$ \frac{c}{-2}=-11 $$

Short Answer

Expert verified
The solution is \( c = 22 \).

Step by step solution

01

Understand the Equation

The equation given is \( \frac{c}{-2} = -11 \). This means that a variable \( c \) is divided by \(-2\), and the result of this division is equal to \(-11\).
02

Eliminate the Fraction

To solve for \( c \), we need to eliminate the division by \(-2\). Multiply both sides of the equation by \(-2\) to isolate \( c \):\[ \left( \frac{c}{-2} \right) \times (-2) = -11 \times (-2) \]This simplifies to:\[ c = 22 \]
03

Verify the Solution

To ensure that \( c = 22 \) is correct, substitute it back into the original equation and check if both sides equal:\[ \frac{22}{-2} = -11 \]Calculate the left side:\[ 22 \div (-2) = -11 \]As both sides are equal, \( c = 22 \) is indeed the correct solution.

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Key Concepts

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

Division Properties in Algebra
In algebra, understanding division properties is crucial for solving equations. Division is not just separating objects; it's reversing multiplication.
  • When a variable divides by a number, it's the same as multiplying by the reciprocal.
  • A key property of division is that dividing both sides of an equation by the same number does not affect the equality of the equation. This is critical when you have equations with fractions.
  • If you have a fraction like \( \frac{c}{-2} \), you can eliminate the division by multiplying by (-2) on both sides, effectively isolating the variable.
Understanding how division properties work helps you manipulate equations and solve for unknowns efficiently.
Verifying Solutions
Once you've solved an equation, always check your solution. This process guarantees that the solution makes sense.
  • To verify, substitute the solution back into the original equation.
  • Calculate both sides of the equation separately to see if they balance out.
  • If both sides of the equation are equal after substitution, the solution is correct.
In our exercise, after solving \( c = 22 \), we checked by substituting \( 22 \) back into the equation \( \frac{22}{-2} \) to see if it equals \(-11\).This practice not only confirms your answer but also strengthens your understanding and confidence in solving equations.
Eliminating Fractions in Equations
Fractions in equations can be challenging, but they're manageable once you know how to eliminate them.
  • The first step is to identify the fraction and the operation involved 鈥� in our case, the operation is division by \(-2\).
  • To remove the fraction, multiply both sides of the equation by the denominator. This action systematically clears the fraction, simplifying the equation.
  • In our example, multiplying both sides by \(-2\) turns \( \frac{c}{-2} = -11 \) into a simpler form - \( c = 22 \).
Eliminating fractions early in the problem helps streamline your calculations and makes checking your work more straightforward.

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