Write and solve equations that represent diagrams

Learning Objectives Identify unknown quantities and represent them with variables in geometric and other visual diagrams. Translate visual information from diagrams, including lengths, angles, or quantities, into algebraic expressions involving rational numbers. Formulate linear equations by setting equivalent expressions equal, based on properties or relationships shown in diagrams. Solve multi-step linear equations that contain rational coefficients and constants. Interpret the solution of an equation in the context of the original diagram and verify its reasonableness. Apply properties of equality and inverse operations to isolate variables in equations derived from diagrams. Ever looked at a picture and wondered how much of something there was, or how long a side really...

TermDefinitionExample VariableA symbol (usually a letter) used to represent an unknown quantity or value in an algebraic expression or equation.In a diagram, if a side length is unknown, we might label it as 'x' or 'w'. Algebraic ExpressionA combination of numbers, variables, and operation symbols (+, -, ×, ÷) that represents a value. It does not contain an equality sign.If a side is twice another side plus 3 units, it could be represented as `2x + 3`. EquationA mathematical statement that shows two expressions are equal. It always contains an equality sign (=).If the perimeter of a rectangle is 20, and its sides are `x` and `x + 2`, the equation is `2x + 2(x + 2) = 20`. Rational NumberAny number that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are...

Properties of Equality If $a = b$, then $a + c = b + c$ (Addition Property), $a - c = b - c$ (Subtraction Property), $ac = bc$ (Multiplication Property), and $\frac{a}{c} = \frac{b}{c}$ (Division Property, $c \neq 0$). These properties allow you to perform the same operation on both sides of an equation to maintain equality and isolate the variable. Distributive Property $a(b + c) = ab + ac$ Use this property to multiply a term outside parentheses by each term inside the parentheses. This is often needed when expressions in diagrams involve grouped quantities. Combining Like Terms Terms with the same variable raised to the same power can be added or subtracted. For example, $ax + bx = (a+b)x$. Simplify expressions by combining terms that are alike. This reduces the n...