Algebraic Expressions & Var Model

Var Model and Equations

• The var model helps visualize equations with rectangles or vars.
• Analyze all choices in "not belong to the group" questions.
• Equations include an equal sign, while expressions do not.

Solving Equations with Var Model Example

• To solve 2x - 6 = 4 using the var model: represent 2x with a rectangle, place -6 below it, and match with 4 below.
• Combining 4 and 6 yields 10, leading to 2x = 10.
• Divide x into 2 equal parts.
• Since x + x = 2x, split 10 into two equal parts: 5 and 5.
• Therefore, x = 5.

Var Model Descriptions and Algebraic Expressions

• When using the var model for x + 5 = 8, place vars x and 5 at the top and 8 at the bottom.
• A variable is a symbol or letter representing an unknown value in an algebraic expression.

Coefficients

• The literal coefficient refers to letters in a term.
• A numerical coefficient is a number.
• In the expression 3xy, xy is the literal coefficient.

Mathematical Phrases and Algebraic Expressions

• When identifying which mathematical phrase doesn't belong in a group, check if the exponent in each letter is a whole number.
• Polynomials have whole number exponents.

Evaluating Algebraic Expressions

• Substitute a value for each variable to evaluate an algebraic expression.
• Evaluate 2m + 3n by substituting m = 4 and n = 2 to get 2(4) + 3(2) = 8 + 6 = 14.
• To evaluate 2x + 9 when x = 10, compute 2 * 10 + 9, which equals 29.

Translating between Algebraic Expressions and Word Sentences

• The phrase can be a verbal phrase or an algebraic expression.
• "The quotient of twice a number and 11" translates to 2x / 11.
• "3 times a number greater than 8" translates to 3y + 8.
• The algebraic expression x - 10 translates to "10 less than a number."

Properties of Equality (Solving Equations)

• Moving a "+ number" to the other side of an equation requires subtracting it from both sides.
• The addition property of equality involves adding the same number to both sides of the equation.
• The symmetric property of equality allows interchanging the left and right sides of an equation.
• If x = 5, then 5 = x, illustrating the symmetric property.

Solving for a Variable

• A literal equation involves two or more variables, with one variable solved in terms of the others.
• Solve one variable in terms of the other to find a literal equation.
• To solve for length (L) in A = LW, divide both sides by W to get L = A / W.
• To solve for height in A = 1/2 * base * height, first remove 1/2.
• Multiply both sides by 2, resulting in 2A = base * height.

Complex Equations and Problem Solving

• To solve 2m + 3n = y for m, transpose 3n: 2m = y - 3n, then divide by 2: m = (y - 3n) / 2.
• To solve 2 + x = y - 7 in terms of y, transpose -7: 2 + x + 7 = y, simplify: y = x + 9.