Problem 67 - Elementary Algebra

Question

Solve. $$ 2(x-3)-6(2 x+1)=-5(2 x-4) $$

Step-by-Step Solution

Answer

No solution; the equation simplifies to a false statement.

Step 1: Expand the expressions

Start by distributing the numbers outside of the parentheses into the brackets. Remember that distribution means you will multiply the term outside by each term inside the parentheses.\[ 2(x-3) = 2x - 6 \]\[ -6(2x+1) = -12x - 6 \]\[ -5(2x-4) = -10x + 20 \]Thus, the equation becomes \( 2x - 6 - 12x - 6 = -10x + 20 \).

Step 2: Combine like terms

Combine the terms that are similar on each side of the equation. Look for terms that have the same pronumeral (i.e., involving \(x\)) and simplify:\[ (2x - 12x) - (6 + 6) = -10x + 20 \]This simplifies to\[ -10x - 12 = -10x + 20 \]

Step 3: Eliminate the x terms

Notice that the \(-10x\) appears on both sides of the equation. Since they are equal and on both sides, we can cancel them out:\[ -12 = 20 \]

Step 4: Analyze the result

The remaining equation \(-12 = 20\) is not true. This indicates there is no value of \(x\) that can satisfy the equation.

Key Concepts

Distributive Property

The distributive property is an essential algebraic tool that allows you to simplify expressions by multiplying a single term by each term inside a set of parentheses. This helps to "distribute" the multiplication over addition or subtraction present in the expression. In the provided equation, we apply the distributive property as follows:

• For the term \(2(x-3)\), multiply 2 by both \(x\) and \(-3\), resulting in \(2x - 6\).
• For \(-6(2x+1)\), multiply \(-6\) by \(2x\) and \(+1\), giving \(-12x - 6\).
• Lastly, for \(-5(2x-4)\), multiply \(-5\) by \(2x\) and \(-4\), resulting in \(-10x + 20\).

The expansion through distribution is crucial because it transforms a complex expression into a simpler one, where like terms can be easily identified and combined.

Solving Equations

Solving equations is the process of finding the values of variables that make the equation true. We rearrange and simplify the terms to isolate the variable and find its value. In our example, the expanded equation is:\(2x - 6 - 12x - 6 = -10x + 20\).The next step in solving the equation is to combine like terms, simplifying each side of the equation. Notably, we observe that similar terms involving the variable \(x\) appear on both sides:

• Combine \(2x\) and \(-12x\) to get \(-10x\).
• Similarly, add the constant terms \(-6\) and \(-6\) to get \(-12\).

Thus, the equation becomes:\(-10x - 12 = -10x + 20\).By solving these steps, we can identify inconsistencies within the equation that indicate a potential absence of solutions, as seen in this specific problem where no solution exists.

Like Terms

Like terms in algebra are terms that have identical variables raised to the same powers, meaning they can be combined through addition or subtraction. Recognizing and combining like terms is a fundamental skill in solving algebraic equations because it simplifies the equation, making it easier to solve.In the equation provided, after applying the distributive property, terms are combined as follows:

• Terms like \(2x\) and \(-12x\) both contain the variable \(x\). These terms are combined to become \(-10x\).
• Similarly, the constants \(-6\) and another \(-6\) can be combined to \(-12\).

This process highlights the importance of identifying and simplifying like terms in order to reach a clearer and more manageable form of the equation. It allows us to observe the structure of the equation and detect any inconsistencies, such as when all variable terms cancel out, leaving a false statement in this case \(-12 = 20\), leading to the conclusion that there is no solution.