Problem 54 - Elementary Algebra

Question

Solve. $$ -9(x-3)-3 x=-3(4 x+9) $$

Step-by-Step Solution

Answer

No solution; the equation is inconsistent.

Step 1: Distribute the Constants

First, we distribute -9 into (x-3) and -3 into (4x+9): \[-9(x-3)\] becomes \(-9x + 27\), and \(-3(4x+9)\) becomes \(-12x - 27\). Therefore, the equation becomes: \[-9x + 27 - 3x = -12x - 27\]

Step 2: Combine Like Terms

Next, combine like terms on the left side of the equation: \(-9x -3x + 27\) becomes \(-12x + 27\). This simplifies the equation to: \[-12x + 27 = -12x - 27\]

Step 3: Simplify and Isolate the Variable

To isolate \(x\), first add \(12x\) to both sides of the equation to eliminate the \(-12x\) terms: \[-12x + 27 + 12x = -12x - 27 + 12x\]Simplifying, we get: \[27 = -27\]

Step 4: Analyze the Solution

The simplified expression, \(27 = -27\), is a false statement. This implies that there is no solution that will satisfy the original equation, meaning it is inconsistent.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic property used to simplify expressions and equations, specifically when you have terms in parentheses. It involves multiplying a single term by each term inside the parentheses.In mathematics, this is expressed as: \[ a(b + c) = ab + ac \] In the exercise, we used the distributive property to simplify the expressions involving \(-9(x-3)\) and \(-3(4x+9)\).

• \(-9(x-3)\) distributes as \(-9 imes x + (-9) imes (-3) = -9x + 27\)
• \(-3(4x+9)\) distributes as \(-3 imes 4x + (-3) imes 9 = -12x - 27\)

By distributing the constants, we removed the parentheses and simplified the equation to \(-9x + 27 - 3x = -12x - 27\), making it easier to manipulate in the following steps.

Combining Like Terms

Combining like terms is crucial in simplifying algebraic expressions. 'Like terms' are terms that have the exact same variable raised to the same power. You can simplify an expression by adding or subtracting these like terms.In our exercise, we combined the terms on the left side of the equation:

• The like terms \(-9x\) and \(-3x\) were combined by adding them: \(-9x - 3x = -12x\)
• Therefore, the left side of the equation simplifies to \(-12x + 27\)

Simplifying expressions by combining like terms is a common step in solving equations because it reduces the equation to its simplest form before further solving steps, like isolating the variable, are applied.

Isolating Variables

Isolating the variable is a fundamental algebra technique used to find the variable's value in an equation. The aim is to get the variable alone on one side of the equation, effectively 'isolating' it.In the original equation, after combining like terms, the equation was \(-12x + 27 = -12x - 27\). One way to isolate terms involving the variable is by trying to eliminate the same terms on both sides.Here's how it was applied:

• We added \(12x\) to both sides of the equation: \(-12x + 27 + 12x = -12x - 27 + 12x\)
• Simplified, it leaves: \(27 = -27\)

The result, \(27 = -27\), indicates something important—since it's a false statement, the original equation has no possible solutions with these variable settings, meaning it is inconsistent. Understanding this process helps in recognizing problems that don't have solutions.