Problem 24 - Prealgebra

Question

Solve each equation. Check your solution. $$86=-2 v$$

Step-by-Step Solution

Answer

The solution is \( v = -43 \).

Step 1: Identify the Equation

The equation given to solve is \( 86 = -2v \). Our goal is to find the value of \( v \).

Step 2: Isolate the Variable

To solve for \( v \), we need to isolate it on one side of the equation. This can be done by dividing both sides by \(-2\) to cancel the coefficient of \( v \).

Step 3: Perform the Division

Divide both sides of the equation by \(-2\) to solve for \( v \): \[ v = \frac{86}{-2} \] Calculate the division: \[ v = -43 \]

Step 4: Check the Solution

To verify the solution, substitute \( v = -43 \) back into the original equation.The original equation is \( 86 = -2(-43) \).Calculate the right-hand side: \[ -2(-43) = 86 \]Both sides equal, confirming \( v = -43 \) is correct.

Key Concepts

Understanding Prealgebra

Prealgebra forms the essential foundation for understanding algebraic concepts. It is like learning the alphabet before forming words.
In this stage, equations are simplified, and focus is primarily on solving basic arithmetic and fundamental equations. This provides a groundwork for more complex mathematics later on.

• Equations: These are statements indicating two expressions are equal, essentially like balanced scales.
• Expressions: Combinations of numbers, variables, and operations but without an equality sign.
• Variables: Symbols, typically letters, that represent unknown values we aim to determine.

Prealgebra prepares students to handle more complicated algebraic operations by ensuring familiarity with numbers, operations, and the basic rules of algebra.

Isolating the Variable

Isolating the variable means getting the variable on one side of the equation to discover its value. This involves basic operations to both sides of the equation, maintaining the balance.
This concept helps in solving for the variable in any equation effectively.

• Identify the variable you need to solve for. Here, it is \( v \).
• Perform inverse operations: Each operation should offset the existing operation, helping you isolate the variable. For instance, if a variable has been multiplied by a coefficient, you divide.

In the provided equation, \( 86 = -2v \), division is used. By dividing both sides by -2, the equation becomes \( v = -43 \). This simple technique is crucial for variable isolation in equations.

Checking Your Solutions

Verifying your solution is essential after solving any algebraic equation. This step ensures that the solution you've found actually satisfies the original equation. It’s a crucial practice for accuracy and confidence in mathematics.
Here's how to check:

• Substitute the known value back into the original equation: Replace the variable with your solution.
• Calculate both sides of the equation to ensure they're equal.

In our example, substituting \( v = -43 \) into \( 86 = -2v \) reaffirms the solution as \(-2(-43) = 86\), confirming the validity of \( v = -43 \). Checking solutions helps avoid mistakes and cement your understanding of the problem.