Problem 68 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ -2 x=5 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -\frac{5}{2}\) or \(x = -2.5\).

Step 1: Identify the equation

The given equation is \(-2x = 5\). Our goal is to find the value of \(x\).

Step 2: Isolate the variable

To solve for \(x\), we need to isolate it on one side of the equation. Since \(-2x\) means \(-2\) multiplied by \(x\), we'll divide both sides by \(-2\) to cancel the \(-2\) on the left side of the equation.

Step 3: Perform the division

Divide both sides by \(-2\) to solve for \(x\): \[x = \frac{5}{-2}\].

Step 4: Simplify the expression

The right side of our division simplifies to \(x = -\frac{5}{2}\) or equivalently, \(x = -2.5\).

Key Concepts

Isolation of Variable

When it comes to solving linear equations, the isolation of the variable is an essential step. The variable is what we are trying to find, and it is often hidden by other numbers or operations in the equation.
In our example, the variable is \( x \), which is being multiplied by \(-2\) in the equation \(-2x = 5\). The strategy here is to get \( x \) by itself on one side of the equation.
We usually begin by looking for opportunities to reverse or "undo" operations impeding the isolation. This often means doing the opposite of whatever operation is affecting the variable. For multiplication, as in our case, division is used. The key is to apply this reversal uniformly to both sides of the equation to maintain its balance. This balance is what ensures the equality holds true.
Breaking it down:

• Identify the operation: Here, it is multiplication by \(-2\).
• Undo the operation by performing the opposite, which is division.

By meticulously doing this, we ensure that the variable stands alone, making it simple to identify its value.

Division in Equations

Division in equations is often used as a tool to simplify an expression or equation. When a variable is multiplied by a number, division helps peel away that layer to find the variable’s value. In our equation, \(-2x = 5\), the \(-2\) is what needs to be removed from the \( x \) to isolate it.
Dividing both sides of the equation by \(-2\) clears the path to finding \( x \). It is crucial to remember that whatever operation you perform on one side of the equation, you must do that same operation to the other side. This preserves the equation's symmetry and accuracy.
Here's how the division works:

• Divide \(-2x\) by \(-2\) which gives \( x \) on the left side.
• Perform the same division on the right side, \( 5 \div (-2) \), to maintain the balance.

Through this process, the equation simplifies, making way for a neat expression where \( x \) is isolated and ready to reveal its value.

Simplifying Expressions

Simplifying expressions makes it easier to interpret and find solutions in equations. In the previous steps, after isolating \( x \) and completing the division, we now have \( x = \frac{5}{-2} \). This expression can be further simplified for clarity.
Simplification is about making an expression cleaner and more understandable without altering its value. Our fraction \( \frac{5}{-2} \) can be expressed as a decimal, which is often more intuitively understood, especially in everyday math situations.
To simplify:

• Recognize the fraction form \( x = \frac{-5}{2} \) or \( x = -\frac{5}{2} \).
• Convert it into a decimal by performing the division: \(-\frac{5}{2} = -2.5\).

This simplification helps put the solution in a more common format, which is easier to comprehend at a glance, showing us clearly that \( x = -2.5 \). Understanding simplification enhances problem-solving skills by allowing clearer communication of answers.