Hey there! 👋 It's totally understandable that linear equations involving fractions and decimals can feel a bit intimidating. Many students find these tricky due to the extra steps. But don't worry, with a few key strategies, they become much more manageable! Let's dive into some practice and tips to build your confidence.
Tackling Linear Equations with Fractions
When you see fractions, your main goal is to "clear" them. This means getting rid of the denominators to leave you with a simpler, integer-only equation. The best way is to find the Least Common Denominator (LCD) of all fractions.
Strategy for Fractions: Multiply every term in the equation by the LCD.
Example 1: Solve for $x$ in the equation: $\frac{x}{3} + \frac{1}{2} = \frac{5}{6}$
Step-by-Step Solution:
- Identify denominators: 3, 2, and 6.
- Find the LCD: The LCD of 3, 2, and 6 is 6.
- Multiply every term by the LCD:
$6 \left( \frac{x}{3} \right) + 6 \left( \frac{1}{2} \right) = 6 \left( \frac{5}{6} \right)$
$2x + 3 = 5$
- Solve the integer equation:
$2x = 5 - 3$
$2x = 2$
$x = 1$
See? Clearing fractions makes it straightforward! ✨
Conquering Linear Equations with Decimals
Decimals can be annoying, especially with mental math or avoiding calculator errors. Similar to fractions, we have a trick to "clear" them and convert them into whole numbers!
Strategy for Decimals: Multiply every term in the equation by a power of 10 (10, 100, 1000, etc.) that will make all decimals integers. Choose the highest number of decimal places in any term to determine the power of 10.
Example 2: Solve for $y$ in the equation: $0.4y - 1.2 = 0.08y + 2.4$
Step-by-Step Solution:
- Identify decimal places: $0.4y$ (one), $1.2$ (one), $0.08y$ (two), $2.4$ (one).
- Determine highest decimal places: Two (from $0.08y$). So, multiply by $10^2 = 100$.
- Multiply every term by 100:
$100(0.4y) - 100(1.2) = 100(0.08y) + 100(2.4)$
$40y - 120 = 8y + 240$
- Solve the integer equation:
$40y - 8y = 240 + 120$
$32y = 360$
$y = \frac{360}{32}$
$y = \frac{45}{4}$ or $y = 11.25$
Practice these strategies, and these equations will become less daunting. The key is to simplify first by clearing those fractions or decimals. You got this! 🚀
