Problem 47 - Elementary Algebra

Question

What is wrong with the following procedure, and how should it be changed? $$ \begin{aligned} 1.2 x+2 &=3.8 \\ 10(1.2 x)+2 &=10(3.8) \\ 12 x+2 &=38 \\ 12 x &=36 \\ x &=3 \end{aligned} $$

Step-by-Step Solution

Answer

The step \(10(1.2x) + 2 = 10(3.8)\) is incorrect; it should be \(10(1.2x + 2) = 38\). Correct solution: \(x = 1.5\).

Step 1: Understanding the Procedure

The given procedure involves solving a linear equation. The equation given is \(1.2x + 2 = 3.8\). The steps attempt to clear a decimal by multiplying through by 10, leading to the transformation of the equation.

Step 2: Identifying the Mistake

The mistake occurs in the second step, \(10(1.2x) + 2 = 10(3.8)\). Only \(1.2x\) was multiplied by 10 instead of multiplying the entire equation, including the \(+2\), by 10.

Step 3: Correct Procedure

Multiply the entire equation by 10: \(10(1.2x + 2) = 10(3.8)\). This results in: \(12x + 20 = 38\).

Step 4: Solving the Corrected Equation

Subtract 20 from both sides to get \(12x = 18\). Then divide both sides by 12, resulting in \(x = \frac{18}{12} = 1.5\).

Step 5: Verification

Substitute \(x = 1.5\) back into the original equation to verify: \(1.2(1.5) + 2 = 1.8 + 2 = 3.8\), which confirms our solution is correct.

Key Concepts

Multiplying Equations

When solving linear equations, multiplying each term by a constant can simplify calculations, especially if decimals are involved. The key principle to remember is that you must multiply every part of the equation by the same constant to maintain its balance. If you multiply an equation by, say, 10, it means applying the multiplication across all terms on both sides of the equation.

• Ensure every term on both sides of an equation is multiplied by the constant.
• This might help clear decimals, turning them into whole numbers.
• Maintaining the balance of the equation is crucial, just like weighing scales; both sides need the same treatment.

Consider the example: if you start with an equation like \(1.2x + 2 = 3.8\), and choose to multiply by 10 to clear the decimals, each term needs to be multiplied. So the correct multiplication action would be: \(10(1.2x + 2) = 10 \times 3.8\), which correctly results in \(12x + 20 = 38\). Remember, failing to multiply a term (like the +2) leads to error.

Decimal Operations

Working with decimals in mathematical equations requires special care to ensure accuracy and simplification. Decimals often appear in linear equations and can sometimes complicate the process of solving them. However, strategic operations can be used to clear decimals for a more straightforward solution.

• Convert decimals to whole numbers for simpler calculations.
• To do this, determine a power of 10 that can convert all decimals in the equation to whole numbers when you multiply the entire equation by it.
• For instance, multiplying by 10 could be helpful if you have one decimal place (e.g., \(1.2\) becomes \(12\)).

Decimals are just parts of whole numbers. Sometimes they can make equations look complex, but with multiplication, they can be transformed into easier-to-handle numbers. For example, in the corrected solution step discussed, multiplying by 10 turns \(1.2x + 2 = 3.8\) into an equation without decimals, making it easier to proceed to the next steps.

Solving Equations

Solving linear equations involves a series of operations that simplify the equation to find the value of the unknown variable. The goal is to isolate the variable on one side of the equation to determine its value that satisfies the equation. Here's a broader process on how to approach solving such equations, particularly after clearing any decimals:

• Start by performing necessary arithmetic operations that simplify the equation, such as adding, subtracting, multiplying, or dividing.
• Move constants to one side of the equation to isolate the variable.
• Simplify the equation step-by-step until you achieve a form such as \(x =\) some value.

For example, after correcting the earlier mistake, you end up with \(12x + 20 = 38\). To solve, you would:
1. Subtract 20 from both sides to simplify, resulting in \(12x = 18\).
2. Divide both sides by 12 to isolate \(x\), yielding \(x = \frac{18}{12} = 1.5\).

Finally, always verify your solution by substituting back into the original equation to see if it holds true, ensuring no errors were made during the process.