Solve equations with variables on both sides
It's time for a variable showdown! When letters appear on both sides of an equation, your goal is to be the ultimate organizer. Gather all the variable terms on one side and all the constant terms on the other to find the solution.
Concept
Solving equations with variables on both sides requires a clear strategy to isolate the variable. The key is to systematically move terms across the equal sign. **The Strategy: Variables on One Side, Constants on the Other** 1. **Simplify Each Side:** If possible, use the distributive property and combine like terms on each side of the equation first. 2. **Move Variable Terms:** Use addition or subtraction to move all the variable terms (like '5x' or '-2y') to one side of the equation. A good habit is to move the smaller variable term to avoid negatives. 3. **Move Constant Terms:** Use addition or subtraction to move all the constant terms (plain numbers like '7' or '-3') to the other side. 4. **Solve for the Variable:** Use multiplication or division to solve for the variable. **Example: 5x + 3 = 2x + 15** 1. **Simplify:** Both sides are already simplified. 2. **Move Variables:** The smaller variable term is 2x. Subtract 2x from both sides to move it. * 5x - 2x + 3 = 2x - 2x + 15 * 3x + 3 = 15 3. **Move Constants:** Subtract 3 from both sides. * 3x + 3 - 3 = 15 - 3 * 3x = 12 4. **Solve:** Divide by 3. * x = 4 **Check your answer:** Plug x=4 back into the original equation: 5(4) + 3 = 2(4) + 15 20 + 3 = 8 + 15 23 = 23. It's correct! **Example with Negatives: 7y - 8 = 3y + 12** 1. **Simplify:** Done. 2. **Move Variables:** Subtract 3y from both sides. * 4y - 8 = 12 3. **Move Constants:** Add 8 to both sides. * 4y = 20 4. **Solve:** Divide by 4. * y = 5 **Key Idea:** Think of the equal sign as a river. When a term crosses the river, it must change its sign (or do the inverse operation) to keep the equation balanced. A "+2x" on one side becomes a "-2x" on the other.
Try it
Practice: Solve equations with variables on both sides.