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Solving Linear Equations 1 Like Terms Like terms contain the same variables raised to the same powers. You can combine like terms by adding or subtracting them (this is not true for unlike terms). To combine like terms, add or subtract the numerical coefficients (as appropriate), then multiply the result by the common variable factors. 2 Combining Like Terms Examples of Combining Terms Terms Before Combining After Combining Terms 6x2 + 7x2 13x2 19xy – 30xy – 11xy 13xy2 – 7x2y Can’t be combined (since the terms are not like terms) 3 Combining Like Terms We cannot combine a chicken and a goat and create a Chickengoat It would be unbalanced!!! 4 Combining Like Terms Or a Zonkey??? 5 Using the Distributive Property Example: Find each product by using the distributive property to remove the parentheses. a.) 2(x + y) b.) 7(x + 2y – 5z) c.) – 4(3a – 3b – 10c) a.) 2(x + y) = 2x + 2y b.) 7(x + 2y – 5z) = 7x + 14y – 35z c.) – 4(3a – 3b – 10c) = – 12a + 12b + 40c 6 Using the Distributive Property Remember to keep it balanced by distributing to everyone 7 Addition Property of Equality Addition Property of Equality If a, b, and c are real numbers, then a = b and a + c = b + c are equivalent equations. Example: a.) 8 + z = – 8 8 + (– 8) + z = – 8 + – 8 z = – 16 Add –8 to each side. Simplify both sides. 8 Solving Equations Example: 5(3 + z) – (8z + 9) = – 4z 15 + 5z – 8z – 9 = – 4z 6 – 3z = – 4z Use distributive property. Simplify left side. 6 – 3z + 4z = – 4z + 4z Add 4z to both sides. 6+z=0 Simplify both sides. 6 + (– 6) + z = 0 +( – 6) Add –6 to both sides. z=–6 Simplify both sides. 9 Multiplication Property of Equality Multiplication Property of Equality If a, b, and c are real numbers, then a = b and ac = bc are equivalent equations Example: –y=8 (– 1)(– y) = 8(– 1) y=–8 Multiply both sides by –1. Simplify both sides. 10 Using Both Properties Example: 3z – 1 = 26 3z – 1 + 1 = 26 + 1 3z = 27 3 z 27 3 3 z=9 Add 1 to both sides. Simplify both sides. Divide both sides by 3. Simplify both sides. 11 Solving Linear Equations Solving Linear Equations in One Variable 1) Multiply both sides by the LCD to clear the equation of fractions if they occur. 2) Use the distributive property to remove parentheses if they occur. 3) Simplify each side of the equation by combining all like terms. 4) Get all variable terms on one side and all numbers on the other side by using the addition property of equality. 5) Get the variable alone by using the multiplication property of equality. 6) Check the solution by substituting it into original equation. 12