Problem 53 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The solution incorrectly handled multiplication. The correct solution is \(x = 1.5\).

Step 1: Verify Given Solution Steps

First, examine each step of the provided solution to understand what was done. The initial equation is \(1.2x + 2 = 3.8\). Then, it's multiplied by 10 to clear the decimal, resulting in \(12x + 2 = 38\). Next, they isolated \(12x\) by subtracting 2 from both sides to get \(12x = 36\). Finally, dividing by 12 gives \(x = 3\).

Step 2: Identify the Mistake

Notice that the mistake is subtly introduced in the multiplication step. When the equation \(1.2x + 2 = 3.8\) is multiplied by 10 to eliminate decimals, it is done correctly to \(12x + 20 = 38\), not \(12x + 2 = 38\). The +20 is from the term "2" also being multiplied by 10.

Step 3: Correct the Multiplication and Solve

Redo the multiplication correctly: Multiply the entire equation \(1.2x + 2 = 3.8\) by 10 to obtain \(12x + 20 = 38\). Now subtract 20 from both sides to isolate the term with \(x\), giving \(12x = 18\). Finally, divide both sides by 12 to solve for \(x\), resulting in \(x = \frac{18}{12} = 1.5\).

Step 4: Verify the Correct Solution

Substitute \(x = 1.5\) back into the original equation to check your work: \(1.2(1.5) + 2\). This simplifies to \(1.8 + 2 = 3.8\), which confirms the solution is correct.

Key Concepts

Decimal Multiplication in Equations

Decimals can make equations look more complex, but they can be simplified by multiplying to convert them into integers. In our problem, the original equation contained decimals: 1.2. To remove these decimals, we multiply the entire equation by 10.

• Multiplying eliminates the decimal, making it easier to solve.
• Ensure every term is multiplied, not just the one with the decimal.

For example, multiplying the entire equation by 10 transforms it to: 12, + 20x = 38. Notice both 1.2 x and the constant 2 were multiplied by 10. This technique simplifies calculations and reduces errors in further steps.

Equation Verification

Verification is the key to ensuring that your solution is correct. Once you identify a potential solution, substitute it back into the original equation. This checks if both sides of the equation remain equal.

• Substitute the solution: Let's check the solution x = 1.5 into the original equation.
• Calculate each side separately: The left side, 1.2(1.5) + 2, equals 3.8.
• The right side of the equation is also 3.8.

Since both sides of the equation are balanced, x = 1.5 is the correct solution. Verification helps catch any potential mistakes made while solving.

Fraction Simplification

When solving for x, you may end up with a fraction. Simplifying fractions is crucial for clarity and understanding. During simplification, fractions are reduced to their simplest form.

• To simplify, find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and denominator by this GCD.

For instance, in the problem, the fraction \(\frac{18}{12}\) can be simplified: * The GCD of 18 and 12 is 6.* Dividing both by 6 gives \(\frac{3}{2}\).Simplifying fractions not only makes them easier to understand but also minimizes errors in further calculations.