Problem 37 - Algebra 1

Question

Solve the equation if possible. $$ -2(6-10 n)=10(2 n-6) $$

Step-by-Step Solution

Answer

The equation has no solution.

Step 1: Distribute the constants

Distribute \(-2\) into the parentheses on the left side to get \(-2*6 + (-2)*(-10n) = -12+20n\). Similarly, distribute \(10\) into the parentheses on the right side to get \(10*2n - 10*6 = 20n-60\). The equation becomes \(-12+20n = 20n-60\).

Step 2: Combine like terms

Combine like terms by subtracting \(20n\) from both sides of the equation. This gives \(-12 = -60\).

Step 3: Evaluate the equality

The equation \(-12 = -60\) is clearly false. This disregard implies there's no \( n \) that will satisfy the original equation.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions by multiplying each term inside a parenthesis by a factor outside it. This property is essential in solving linear equations like the one in our example:
\( -2(6-10n) = 10(2n-6) \).
Understanding the distributive property helps break down complicated expressions into manageable parts.
For instance, by distributing the

• negative sign and the number -2 across both terms inside the parenthesis, we change \( -2(6-10n) \) to \( -2 \times 6 + (-2) \times (-10n) \), which simplifies to \(-12 + 20n\)
• On the right side, distributing 10 throughout \( (2n-6) \) results in \( 10*2n - 10*6 \), which simplifies to \( 20n-60 \).

These steps are crucial as they transform the equation into a form where other operations can be applied, eventually solving for the variable. It's important to apply the distributive property correctly to avoid errors in further steps.

Combining Like Terms

After using the distributive property, we often need to combine like terms to create simpler expressions. Like terms have identical variable parts, which means they can be added or subtracted. In our exercise, after distribution, we had the equation:
\(-12+20n = 20n-60\).
By observing both sides, you'll notice the term \(20n\) appears in both parts of the equation:

• To make the equation simpler, we subtract \(20n\) from both sides.
• This operation eliminates \(20n\) on both sides, helping to isolate other terms for further analysis.

This process of combining like terms reduces complexity, bringing us closer to understanding the equation's solution—or lack thereof. In this scenario, combining terms correctly reveals that the remaining numbers lead to a false statement.

No Solution Equations

Not all linear equations have a solution, and sometimes you may end up with a statement that doesn't make sense numerically. This indicates a 'no solution' situation, which is exactly what happened in our original problem.
After distributing and combining like terms, the equation became:
\(-12 = -60\).
Since -12 does not equal -60, the reality here is that no value of \( n \) will satisfy the original equation.

• Such outcomes hint at equations that are structurally contradictory.
• In practical terms, they express conditions that cannot logically coexist.

Recognizing a 'no solution' equation is vital because it saves you effort in trying to find a nonexistent result. In exams or homework, always check your final equality to confirm whether a variable can actually satisfy the equation's requirements.