Equations as Balance

Objective: Distinguish statements from expressions and test solutions.

An equation asserts that two expressions are equal. A solution is a value in the permitted domain that makes the equality true. For 2x+3=11, x=4 is a solution because 2(4)+3=11; x=3 is not. An expression such as 2x+3 has no equality sign and is not itself a statement to solve. A formula is an equation describing a relationship among quantities. Some equations are true for one value, all values in a domain, or no values; always test candidate values in the original equation, especially after transformations that may change the domain.

Objective: Preserve equality with the same operation on both sides.

If a=b, then a+c=b+c and a−c=b−c. Multiplying both sides by the same number preserves equality; division by the same nonzero number also does. These properties justify inverse operations in equation solving. For x+7=12, subtract seven from both sides to get x=5. Dividing by zero is prohibited and multiplying by zero may erase information, turning different original statements into 0=0; such steps are not reversible. A valid solution method should maintain equivalence and check the final candidate.

Objective: Define a variable, write a model, and interpret it.

Let the variable denote a quantity with units, translate the relationship into an equation, solve, and check against both arithmetic and context. If a total fare is a fixed $4 fee plus $3 per mile and the fare is $19, then 4+3m=19 gives m=5 miles. The algebraic answer must respect the domain m≥0; a negative number of miles would be inappropriate here. If five identical items cost $35 with no other charges, 5p=35 implies p=$7 per item. These models assume the stated rate and fee apply as given and omit tax or other charges unless included.