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3.1 Linear Equations: One Transformation Linear Equations in One Variable A linear equation in one variable is an equation that can be written in the form ax + by = c, where a, b, and c are real numbers and a ≠ 0. An equation is a statement with an equals sign. 3.1 Linear Equations:One Transformation Original Equation Equivalent Equation 1. Add the same number to both sides. x-4=7 x=3 2. Subtract the same number to both sides. x+2=9 x=7 3. Multiply both sides by the same number. x/2 = 4 x=8 4. Divide both sides by the same number. 5x = 20 x=4 x=7 7=x 5. Interchange the two sides. 3.1 Linear Equations:One Transformation Two equations are equivalent if they have the same solution set. In solving an equation, the goal is to isolate the variable by using inverse operations. 3.1 Linear Equations:One Transformation Solve x+3=7 x + 3 - 3 = 7 - 3 Add -3 to both sides x=4 Simplify 3.1 Linear Equations:One Transformation Solve 3x = -12 3x 12 Divide both sides by 3 3 3 x 4 Simplify 3.1 Linear Equations:One Transformation Solve -4 = x - 7 4 +7 = x - 7 +7 Add 7 to both sides. 3 x Simplify x3 Transpose 3.1 Linear Equations:One Transformation Solve x 5 x 5 = -9 5 = -9(5) Multiply both sides by 5. x 45 Simplify 3.1 Linear Equations:One Transformation Properties of Equalities 1. If a = b, then a + c, = b + c Addition Property 2. If a = b, then a - c, = b - c Subtraction Property 3. If a = b, then ac = bc Multiplication Property a b 4. If a b and c 0, then Division Property c c 3.2 Linear Equations: Two or More Transformations Using Two or More Transformations 1. Simplify both sides of the equation. 2. Use inverse operations to isolate the variable. 3. Check the solution. 3.2 Linear Equations:Two or More Transformations Solve 4 x + 7 - 2x =13 2x 7 13 Combine like terms. 2x 6 Subtract 7 from both sides. x3 Divide both sides by 2. 3.2 Linear Equations:Two or More Transformations Solve 5x +2(3 - x) =15 5x 6 2x 15 Remove parentheses 3x 6 15 Combine like terms. 3x 9 x3 Subtract 6 from both sides. Divide both sides by 3. 3.2 Linear Equations: Two or More Transformations Solve: 1. 3x + 6 = 12 2. 4.5 = 3 + 2x 3. -2x + 2 - 4x = 20 4. 4(x - 2) = -10 x x 1 5. 6 8 8 3.3 Solving Equations with Variables on Both Sides Collect variables on the side with the greatest variable coefficient 3x 5 8x 30 3x 5 8x 30 3x 5 3x 8x 30 3x 3x 5 8x 8x 30 8x 5 5x 30 35 5x 7 x 5x 5 30 5x 35 x 7 3.3 Solving Equations with Variables on Both Sides Solve: 4(y 2) 6y 2 8y 4 y 8 2y 2 4 y 8 2y 2y 2 2y 6y 8 2 6y 8 8 2 8 6y 6 y 1 3.3 Solving Equations with Variables on Both Sides Solve: 3(x 5) 2x 10 4x 3x 15 2x 10 5x 15 10 5x 5 x 1 3.4 Problem Solving General Strategies for Problem Solving 1. Read and understand the problem. Choose a variable Construct a diagram 2. Translate the problem into an equation. 3. Solve the equation. 4. Interpret the results.(Check your answer) 3.5 Solving Equations That Involve Decimals Solve: 12.3x - 5.1 = 17 12.3x - 5.1+5.1 = 17 + 5.1 12.3x = 22.1 22.1 x 12.3 x 1.796747 3.6 Literal Equations Solve for v d =vt d v t Solve for x y = mx + b y b mx y b x m Solve for l 2w +2l =P 2l =P - 2w P 2w l 2 3.7 Scatter Plots Quadrant II Quadrant I Quadrant III Quadrant IV 3.7 Scatter Plots A(3,2) A B(-1,-3) B C(-5,0) B