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Algebra 1 Notes SOL A.4 Equations Mr. Hannam Name: __________________________________________ Date: _______________ Block: _______ Equations An equation is an open sentence where two expressions are set _____________. Equations may have one or more unknowns (_______________) which we may try to solve. Solving an equation means finding the value(s) of variable(s) that make the equation ____________. Equations that have the same solution(s) are called ____________________. We use ______________________ operations to solve equations. We justify the steps in solving equations by using field properties. Solving Equations “Isolate” the variable, justifying steps using field properties (properties of equality): 1) Put variables on one side of the = and numbers on the other by isolating x: perform inverse operations (add, subtract, multiply, or divide) 2) Whatever you do to one side of the equation, you do to the other 3) Verify solutions S substitute solution in original equation (DO NOT SKIP THIS STEP!!!!!!) Field Properties of Equality: Property of Equality Reflexive Property of Equality Algebra (for real numbers a, b, c) a=a Symmetric Property of Equality If a = b, then b = a Transitive Property of Equality If a = b and b = c, then a = c Substitution Property of Equality If a = b, then a can be substituted for b Addition Property of Equality If a = b, then a + c = b + c Subtraction Property of Equality If a = b, then a – c = b - c Multiplication Property of Equality If a = b, then ac = bc Division Property of Equality If a = b and c 0, then a b c c Example Algebra 1 Notes SOL A.4 Equations Solve the equations: a) x – 4 = 6 x–4+4=6+4 x = 10 VERIFY: Is 10 a solution? (10) – 4 = 6 6=6 Mr. Hannam Page 2 b) x + 3 = -5 x + 3 – 3 = -5 – 3 x = -8 VERIFY: Is -8 a solution? (-8) + 3 = -5 -5 = -5 c) 4x = 16 4x ÷ 4 = 16 ÷ 4 x=4 VERIFY: Is 4 a solution? 4(4) = 16 16 = 16 Solve the equations, justifying steps: a) Solve x + 4 = 10 1) x + 4 = 10 2) x + 4 – 4 = 10 – 4 3) x = 6 b) Solve Given Subtraction Property of Equality Simplify 2 x=4 3 2 x=4 Given 3 3 2 3 x 4 Mult. Property of Equality 2) 2 3 2 3) x = 6 Simplify 1) Two-Step Equations SAME! Put variables on one side of the = and numbers on the other: How to Solve Two-Step Equations 1) Clear parentheses (distribute) and combine like terms if necessary 2) Do add/subtract steps first 3) Do multiply/divide steps Examples: x 5 11 2 x 5 5 11 5 ________________ 2 x 16 ________________ 2 x 2 2 16 ________________ 2 x = 32 _________________ VERIFY: a) 5x + 9 = 24 5x + 9 – 9 = 24 – 9 __________________ 5x = 15 __________________ 5x 15 __________________ 5 5 x=3 __________________ VERIFY: b) c) 3x + 2x = 15 d) 4(x - 6) = 32 VERIFY: VERIFY: Algebra 1 Notes SOL A.4 Equations Mr. Hannam Page 3 Practice - solve the equations, justifying steps with field properties: a) x + 9 = 25 b) -4x = -20 a 46 3 e) f) -1 = z 3 37 3 x 3 5 d) 2x + 3 = -9 g) 7x – 4x = 21 h) 3(x + 2) = 9 c) Multistep Equations and Equations with Variables on Both Sides How to Solve Multi-Step Equations and Equations with Variables on Both Sides 1) Clear parentheses (distribute) and combine like terms if necessary 2) Add/subtract variable terms so that variable is on one side (NEW STEP!) 3) Do add/subtract steps first 4) Do multiply/divide steps Example: Solve 7 – 3x = 4x – 7 1) 7 – 3x = 4x – 7 Given 2) 7 – 3x - 4x = 4x – 7 - 4x ________________________________________________ 3) 7 – 7x = - 7 ________________________________________________ 4) 7 -7x - 7 = -7 - 7 ________________________________________________ 5) -7x = -14 ________________________________________________ 6) ________________________________________________ - 7x - 14 7 -7 7) x = 2 ________________________________________________ Examples: Solve the equation, justifying steps… 1 a) 9x – 5 = (16x + 60) b) 3(x + 12) – x = 4(2 – x) – 3x + 2x 4 9x – 5 = 4x + 15 ______________________ 5x – 5 = 15 ______________________ 5x = 20 ______________________ x=4 _____________________ Algebra 1 Notes SOL A.4 Equations Mr. Hannam Page 4 Special cases: Identity case (solutions are all real numbers): No solution case: 2x + 6 = 2(x + 3) 3x = 3(x + 4) 2x + 6 = 2x + 6 Distribute 3x = 3x + 12 Distribute 0=0 Subtraction Prop. of = 0 = 12 Subtraction Prop. of = Where did the variable go?? Where did the variable go?? When we get a TRUE statement at the end when the variable “disappears,” EVERY x is a solution (the two sides of the equation are identical!). When we get a FALSE statement at the end when the variable “disappears,” there are NO SOLUTIONS. There is no value of x that makes the equation true. ALL REAL NUMBERS are solutions. Other Equations: Solve: a) x 3 4 2 Cross Product Property: b) If x 2 x 1 3 a c then ad = bc b d REMEMBER TO GROUP NUMERATORS AND DENOMINATORS (use parentheses) You Try: Solve the equation if possible; if not, write “all real numbers” or “no solution”… a) 8x + 5 = 6x + 1 b) x + 1 = 3x - 1 c) 9x = 6(x + 4) d) 7(x + 7) = 5x + 59 e) 2 – 15x = 5(-3x + 2) f) 12y + 6 = 6(2y + 1) h) 40 + 14x = 2(-4x – 13) i) 2(3x + 2) = g) 5(x + 2) = j) x 9 2 3 3 (5 + 10x) 5 k) 4 8 x -8 2 l) 2 x 5 21 - x 1 (12x + 8) 2