Solving Equations
Solving equations is a core algebra skill in Grade 8 math (Yoshiwara Core Math) that finds the value of an unknown variable making an equation true. The strategy: isolate the variable by applying inverse operations to both sides — subtraction to undo addition, division to undo multiplication. For example, 3x + 5 = 20: subtract 5 from both sides (3x = 15), then divide by 3 (x = 5). Every step must maintain equality on both sides. This skill underlies all of algebra, function analysis, and real-world formula application.
Key Concepts
Property We solve an equation by undoing in reverse order the operations performed on the variable. We can think of any string of terms as a sum, where the $+$ and $ $ symbols tell us the sign of the term that follows. To solve, first add or subtract terms to isolate the variable term, then multiply or divide to find the variable's value.
Examples To solve $12 4x = 8$, first subtract 12 from both sides to get $ 4x = 20$. Then divide by $ 4$ to find $x = 5$. To solve $ 3y + 7 = 22$, first subtract 7 from both sides to get $ 3y = 15$. Then divide by $ 3$ to get $y = 5$. To solve $\frac{x 5}{3} = 2$, first multiply both sides by 3 to get $x 5 = 6$. Then add 5 to both sides to find $x = 1$.
Explanation Solving an equation is like being a detective to find the variable's hidden value. You reverse the equation's steps, undoing each operation one by one until the variable is alone on one side of the equals sign.
Common Questions
What does it mean to solve an equation?
Finding the variable value that makes both sides equal. This value is the solution or root.
What are inverse operations in equation solving?
Operations that undo each other: addition/subtraction are inverses; multiplication/division are inverses.
How do you solve 3x + 5 = 20?
Subtract 5 from both sides: 3x = 15. Divide both sides by 3: x = 5.
Why apply the same operation to both sides?
An equation is a balance. Applying the same operation to both sides maintains equality.
How do you check if your solution is correct?
Substitute back into the original equation. If both sides are equal, the solution is correct.