Problem 38 - Elementary Algebra

Question

Solve. $$ -2 y-5=8 y-6-10 y $$

Step-by-Step Solution

Answer

No solution, as the equation is inconsistent.

Step 1: Simplify the Equation

Start by simplifying the terms on the right side of the equation. Combine the like terms related to "y": \[-2y - 5 = 8y - 6 - 10y\] Which simplifies to: \[-2y - 5 = -2y - 6\]

Step 2: Eliminate the Variable Through Addition

To eliminate the variable "y" on both sides, add \(2y\) to each side of the equation: \[-2y + 2y - 5 = -2y + 2y - 6\]This simplifies to: \[-5 = -6\]

Step 3: Analyze the Simplified Equation

After simplifying, we reach an equation where the numerical values are not equal, i.e., \(-5 = -6\). This indicates that there is no valid solution, as the equation leads to a contradiction.

Key Concepts

Understanding Like Terms

In a linear equation, like terms are terms that have the same variable raised to the same power. For example, in the equation \(-2y - 5 = 8y - 6 - 10y\), the terms \(-2y\), \(8y\), and \(-10y\) are like terms because they all involve the variable \(y\) raised to the first power.
Combining like terms helps to simplify the equation, making it easier to solve. In our initial equation; combining - \(8y\) and \(-10y\) gives \(-2y\),
So, the equation simplifies to \(-2y - 5 = -2y - 6\).
Simplifying by combining like terms is the first crucial step in solving linear equations.

Recognizing a Contradiction

A contradiction in an equation occurs when the simplified version of an equation results in a statement that is clearly false, such as \(-5 = -6\).
This happens when the process of simplifying leads distinct numerical values on opposite sides of the equation that cannot possibly be equal.
In our example, after eliminating the variable by adding \(2y\) to both sides, we ended up with this contradiction.

• It suggests an inherent inconsistency within the equation.
• Such outcomes typically emerge from equations formed through specific combinations of terms that cannot satisfy equality.

Recognizing a contradiction is crucial as it informs us that there is a fundamental flaw in the logic of the equation.

The Notion of No Solution

When an equation results in a contradiction, as seen with \(-5 = -6\), it means that there is no solution.
This is because there is no possible value for the variable that will satisfy the equation.
The concept of 'no solution' is fundamental in mathematics as it helps to understand when an equation is unsolvable given the conditions presented.

• No possible substitution can make both sides of the equation equal.
• Such equations often, though not always, arise from transformed solutions that were manipulated incorrectly or constraints where no common set of values is applicable.

Understanding when there is no solution is vital, as it helps identify errors in setting up or interpreting equations.