Isolating the variable $x$ in an equation is a fundamental skill in algebra. The goal is to get $x$ by itself on one side of the equation. Let’s walk through the steps with some examples.
Step-by-Step Process
1. Simplify Both Sides
First, simplify both sides of the equation by combining like terms and eliminating any unnecessary parentheses.
Example:
$5x + 3 – 2x = 12 + 4$
Simplify to:
$3x + 3 = 16$
2. Move Variable Terms to One Side
Next, move all terms containing $x$ to one side of the equation and constant terms to the other side. Use addition or subtraction to achieve this.
Example:
$3x + 3 = 16$
Subtract 3 from both sides:
$3x = 13$
3. Isolate the Variable
To isolate $x$, divide both sides of the equation by the coefficient of $x$
Example:
$3x = 13$
Divide by 3:
$x = frac{13}{3}$
4. Check Your Solution
It’s always a good idea to check your solution by substituting $x$ back into the original equation to ensure it balances.
Example:
Substitute $x = frac{13}{3}$ back into the original equation $5x + 3 – 2x = 16$:
$5 times frac{13}{3} + 3 – 2 times frac{13}{3} = 16$
Simplify to check:
$16 = 16$
Special Cases
Equations with Fractions
If the equation contains fractions, first clear the fractions by multiplying every term by the least common denominator (LCD).
Example:
$frac{2x}{3} + 5 = frac{x}{2} – 1$
Multiply every term by 6 (LCD of 3 and 2):
$4x + 30 = 3x – 6$
Then, follow the usual steps to isolate $x$:
Subtract $3x$ from both sides:
$x + 30 = -6$
Subtract 30 from both sides:
$x = -36$
Equations with Parentheses
If an equation has parentheses, first use the distributive property to eliminate them.
Example:
$3(x + 4) = 18$
Distribute 3:
$3x + 12 = 18$
Then, follow the usual steps:
Subtract 12:
$3x = 6$
Divide by 3:
$x = 2$
Conclusion
Isolating $x$ involves simplifying the equation, moving terms to one side, and using basic arithmetic operations. Practice these steps to become proficient in solving for $x$ in various types of equations.