Linear Equations in One Variable 6.2 Linear Equations in One Variable

Definitions Terms are parts that are added or subtracted in an algebraic expression. Coefficient is the numerical part of a term. Like terms are terms that have the same variables with the same exponents on the variables. Unlike terms have different variables or different exponents on the variables.

Properties of the Real Numbers Associative property of multiplication (ab)c = a(bc) Associative property of addition (a + b) + c = a + (b + c) Commutative property of multiplication ab = ba Commutative property of addition a + b = b + a Distributive property a(b + c) = ab + ac

Example: Combine Like Terms 8x + 4x = (8 + 4)x = 12x 5y  6y = (5  6)y = y x + 15  5x + 9 = (1 5)x + (15 + 9) = 4x + 24 3x + 2 + 6y  4 + 7x = (3 + 7)x + 6y + (2  4) = 10x + 6y  2

Practice Problems: p. 306 # 22

Solving Equations Addition Property of Equality If a = b, then a + c = b + c for all real numbers a, b, and c. Find the solution to the equation x  9 = 24. x  9 + 9 = 24 + 9 x = 33 Check: x  9 = 24 33  9 = 24 ? 24 = 24 true

Solving Equations continued Subtraction Property of Equality If a = b, then a  c = b  c for all real numbers a, b, and c. Find the solution to the equation x + 12 = 31. x + 12  12 = 31  12 x = 19 Check: x + 12 = 31 19 + 12 = 31 ? 31 = 31 true

Solving Equations continued Multiplication Property of Equality If a = b, then a • c = b • c for all real numbers a, b, and c, where c  0. Find the solution to the equation

Solving Equations continued Division Property of Equality If a = b, then for all real numbers a, b, and c, c  0. Find the solution to the equation 4x = 48.

General Procedure for Solving Linear Equations If the equation contains fractions, multiply both sides of the equation by the lowest common denominator (or least common multiple). This step will eliminate all fractions from the equation. Use the distributive property to remove parentheses when necessary. Combine like terms on the same side of the equal sign when possible.

General Procedure for Solving Linear Equations continued Use the addition or subtraction property to collect all terms with a variable on one side of the equal sign and all constants on the other side of the equal sign. It may be necessary to use the addition or subtraction property more than once. This process will eventually result in an equation of the form ax = b, where a and b are real numbers. Solve for the variable using the division or multiplication property. The result will be an answer in the form x = c, where c is a real number.

Example: Solving Equations Solve 3x  4 = 17.

Example: Solving Equations Solve 21 = 6 + 3(x + 2) .

Example: Solving Equations Solve 8x + 3 = 6x + 21.

Example: Solving Equations Solve False, the equation has no solution. The equation is inconsistent.

Example: Solving Equations Solve True, 0 = 0 the solution is all real numbers.

Practice Problems: p. 307 # 48, 50 & 58

Proportions A proportion is a statement of equality between two ratios. Cross Multiplication If then ad = bc, b  0, d  0.

To Solve Application Problems Using Proportions Represent the unknown quantity by a variable. Set up the proportion by listing the given ratio on the left-hand side of the equal sign and the unknown and other given quantity on the right-hand side of the equal sign. When setting up the right-hand side of the proportion, the same respective quantities should occupy the same respective positions on the left and right. For example, an acceptable proportion might be

To Solve Application Problems Using Proportions continued Once the proportion is properly written, drop the units and use cross multiplication to solve the equation. Answer the question or questions asked using appropriate units.

Example A 50 pound bag of fertilizer will cover an area of 15,000 ft2. How many pounds are needed to cover an area of 226,000 ft2? 754 pounds of fertilizer would be needed.

Homework P. 306 # 15 – 71 eoo