Problem 77 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$3(2-7 x)=3(4-7 x)$$

Step-by-Step Solution

Answer

The equation \(3(2-7x)=3(4-7x)\) has no solution.

Step 1: Distribute

Using the distributive property to remove parentheses, the given equation \(3(2-7x)=3(4-7x)\) results in \(3*2 - 3*7x = 3*4 - 3*7x\). This simplifies into \(6 - 21x = 12 - 21x\).

Step 2: Subtract 6 from both sides

In order to further simplify and balance the equation, we must subtract 6 from both sides. Doing so gives \(-21x = 6 - 21x\).

Step 3: Eliminate x

If we add \(21x\) to both sides of the equation, we will be left with \(0 = 6\).

Step 4: Contradiction

The statement \(0 = 6\) is a contradiction, thus the equation has no solution.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to remove parentheses by redistributing multiplication over addition or subtraction within the parenthesis. For instance, in the expression \( 3(2 - 7x) \), we apply the distributive property by multiplying each term inside the parentheses by 3, yielding \( 3 \times 2 - 3 \times 7x \), which simplifies to \( 6 - 21x \). It's crucial for students to grasp this property as it's widely used in algebraic problem-solving.

When faced with solving algebraic expressions, first look for opportunities to apply the distributive property to make the rest of the problem simpler. Also, remember that this property is not just about eliminating parentheses but ensuring that every term within the parentheses is correctly multiplied by the number or variable outside.

Equation Simplification

Simplifying equations is all about making them easier to solve. After applying the distributive property, like we did to get \( 6 - 21x = 12 - 21x \), the next step is typically to combine like terms and isolate the variable. In our example, though, we noticed that \(-21x\) appears on both sides of the equation. Subtracting \(6\) from both sides, as shown in Step 2, is a critical simplification step that often leads to making further progress or, as in this case, revealing the nature of the equation.

It's essential to perform simplification steps carefully and methodically. Add or subtract terms on both sides to get variables on one side and constants on the other, and multiply or divide when necessary to isolate the variable. Consistency in performing these actions can prevent mistakes that could lead to the wrong conclusion about an equation's solution.

No Solution Contradiction

Certain algebraic equations lead to contradictions, revealing that they have no solutions. In our example, after simplifying and attempting to isolate the variable \(x\), we ended up with the equation \(0 = 6\), which is an impossible scenario. This contradiction signals that there is no possible value of \(x\) that could make the original equation true, hence the equation is deemed to have no solution.

It's vital to recognize that not every algebraic equation will have a solution. Identifying a contradiction is as important as finding a solution because it represents the correct understanding and analysis of the given problem. Whenever you reach a nonsensical statement, like comparing a nonzero constant to zero, you can confidently conclude that the equation has no solution.

Algebraic Problem-Solving

Algebraic problem-solving encompasses finding solutions to equations by applying a series of reasoned steps. Each step involves specific algebraic principles to manipulate the equation methodically toward a solution. In the example above, problem-solving involved distributing, simplifying, attempting to isolate the variable, and then recognizing the contradiction.

Effective algebraic problem-solving requires practice, a solid understanding of algebraic principles, and careful step-by-step work. There may be times when rearranging the equation or applying a different method is the key to solving it. Always take the time to review each step and ensure your logic is sound, as premature conclusions can lead to misinterpretation. Whether the result is a specific value, a set of values, or a realization that no solutions exist, the goal is to arrive at a justified conclusion based on sound algebraic procedures.