Problem 20 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ -20 x=100 $$

Step-by-Step Solution

Answer

Answer: x = -5

Step 1: Identify the equation

We have the equation: $$ -20x = 100 $$

Step 2: Isolate x

Divide both sides of the equation by -20 to isolate x: $$ \frac{-20x}{-20} = \frac{100}{-20} $$

Step 3: Simplify and solve for x

Now, simplify the equation and solve for x: $$ x = -5 $$ So, the solution for x in this conditional equation is x = -5.

Key Concepts

Understanding Conditional Equations

In algebra, a conditional equation is an equation that holds true for some values of the variable but not for others. For example, the equation \(-20x = 100\) will only be true for particular values of \(x\). Understanding this condition is vital to solving many algebraic equations.

A conditional equation contrasts with identities, where the equation is true for all values of the variable, and contradictions, where there are no possible values that satisfy the equation. When dealing with conditional equations:

• Identify the specific value or values that satisfy the equation.
• Consider that sometimes additional steps are necessary to determine the solution accurately.

By solving the equation, we find that \(x = -5\) works, meaning that is the only number for this particular equation that makes the statement true. Thus, understanding conditionality helps to pinpoint when equations have limited solutions.

Mastering Division in Algebra

Division in algebra is a lot like regular division but with variables. When you divide each side of an equation, you do it to make simplifications and to solve for unknowns. In our example, we use division to eliminate the coefficient of \(-20\) that is attached to \(x\).

Here’s what you should remember when dividing in algebra:

• Always perform the same operation on both sides of the equation to keep it balanced.
• Divide both sides of the equation by the number that is multiplied by the variable to isolate it. In our case, divide by \(-20\).
• Be mindful of negative numbers, as dividing by a negative will change the sign of the result.

If you divide both sides of the equation \(-20x = 100\) by \(-20\), you get \(x = \frac{100}{-20} = -5\). This division technique helps in reducing an equation to a simpler form, making it easier to identify the value of the variable.

Isolation of Variable: A Powerful Technique

Isolation of a variable means rearranging an equation so that the variable stands alone on one side of the equation. This is a fundamental technique used to solve equations and is an essential skill in algebra.

Here's how you achieve the isolation of the variable in an equation:

• Identify the operations that are performed on the variable.
• Perform inverse operations to cancel these operations. In our example, multiplication by \(-20\) is canceled by division.
• Ensure that whatever operation you perform on one side, you also do to the other side of the equation. This maintains the equality.

By isolating the variable, you simplify complex equations into basic arithmetic problems. In the equation \(-20x = 100\), once we divide both sides by \(-20\), we are left with \(x = -5\). This showcases how isolating the variable allows you to solve the equation clearly and correctly.