Algebra I End-of-Course (EOC)

An equation is a mathematical statement showing that two expressions are equal. The goal of solving an equation is to find the value of the unknown variable that makes the statement true. The variable is typically represented by a letter such as x, y, or n.

The fundamental principle of solving equations is maintaining balance. Whatever operation you perform on one side of the equation, you must perform the same operation on the other side. This preserves the equality and allows you to isolate the variable.

One-step equations require only a single operation to isolate the variable. These equations involve one of the four basic operations: addition, subtraction, multiplication, or division.

Addition Equations: When a number is added to the variable, subtract that number from both sides to isolate the variable.

Subtract 7 from both sides: x + 7 - 7 = 12 - 7

Solution: x = 5

To verify, substitute 5 back into the original equation: 5 + 7 = 12. This is true, so x = 5 is correct.

Subtraction Equations: When a number is subtracted from the variable, add that number to both sides.

Solution: x = 13

Multiplication Equations: When the variable is multiplied by a number, divide both sides by that number.

Divide both sides by 5: 5x/5 = 35/5

Solution: x = 7

Division Equations: When the variable is divided by a number, multiply both sides by that number.

Multiply both sides by 4: (x/4) × 4 = 6 × 4

Solution: x = 24

Understanding inverse operations is critical for solving equations. Addition and subtraction are inverse operations. Multiplication and division are inverse operations. To isolate a variable, apply the inverse operation to undo what has been done to the variable.

Two-step equations require two operations to solve. The standard approach is to first undo addition or subtraction, then undo multiplication or division. This follows the reverse order of operations.

Step 1: Add 9 to both sides. 4x = 36

Step 2: Divide both sides by 4. x = 9

When the coefficient of the variable is negative, the same principles apply.

Step 1: Subtract 8 from both sides. -2x = 6

Step 2: Divide both sides by -2. x = -3

Step 1: Add 12 to both sides. 5x = 5

Step 2: Divide both sides by 5. x = 1

Solutions to equations are not always whole numbers.

Step 1: Subtract 3 from both sides. 2x = 7

Step 2: Divide both sides by 2. x = 7/2 or x = 3.5

Performing different operations on each side of the equation destroys the balance and leads to incorrect answers. Forgetting to apply operations to all terms on a side is another frequent mistake. Always verify your solution by substituting it back into the original equation.