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1.4 Solving Linear Equations

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Blitzer, Algebra for College Students, 6e – Slide #2 Section 1.4 Linear Equations Definition of a Linear Equation A linear equation in one variable x is an equation that can be written in the form ax + b = 0, where a and b are real numbers and a is not equal to 0. An example of a linear equation in x is 4x + 2 = 6. Linear equations in x are first degree equations in the variable x.

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Blitzer, Algebra for College Students, 6e – Slide #3 Section 1.4 Solving Linear Equations Solving a Linear Equation 1) Simplify the algebraic expressions on each side. 2) Collect all the variable terms on one side and all the numbers, or constant terms, on the other side 3) Isolate the variable and solve. 4) Check the proposed solution in the original equation.

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Blitzer, Algebra for College Students, 6e – Slide #4 Section 1.4 Solving Linear EquationsEXAMPLE SOLUTION Solve and check: 5 - 3x + 4x = 1 - 7x + 12. 1) Simplify the algebraic expressions on each side. 5 - 3x + 4x = 1 - 7x + 12 5 + x = 13 - 7x 2) Collect variable terms on one side and constant terms on the other side. 5 + x = 13 – 7x + 7x + 7x 5 + 8x = 13

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Blitzer, Algebra for College Students, 6e – Slide #5 Section 1.4 Solving Linear EquationsCONTINUED 5 + 8x = 13Simplify Subtract 5 from both sides 8x = 8 -5 3) Isolate the variable and solve. Divide both sides by 8 8 8 x = 1

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Blitzer, Algebra for College Students, 6e – Slide #6 Section 1.4 Solving Linear Equations 4) Check the proposed solution in the original equation. CONTINUED 5 - 3x + 4x = 2 - 7x + 6Original equation 5 – 3(1) + 4(1) 1 – 7(1) + 12Replace x with 1 ? = 5 – 3 + 4 1 – 7 + 12Multiply= ? 2 + 4 – 6 + 12Add or subtract from left to right = ? Add6 = 6

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Blitzer, Algebra for College Students, 6e – Slide #7 Section 1.4 Solving Linear Equations EXAMPLE #2 SOLUTION Solve and check: 2x – 7 +x = 3x + 1 + 2x.

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Blitzer, Algebra for College Students, 6e – Slide #8 Section 1.4 Solving Linear Equations EXAMPLE #3 SOLUTION Solve and check: 4(2x + 1) – 29 = 3(2x – 5).

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1.4 Assignment (part 1) p. 46 (2-24 even)

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Blitzer, Algebra for College Students, 6e – Slide #10 Section 1.4 Solving Linear Equations EXAMPLE #4 SOLUTION Solve and check: Multiply both sides by the LCD: 6 Distributive Property

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Blitzer, Algebra for College Students, 6e – Slide #11 Section 1.4 Solving Linear Equations Cancel CONTINUED Multiply Combine like terms and solve for x.

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Blitzer, Algebra for College Students, 6e – Slide #12 Section 1.4 Solving Linear Equations EXAMPLE #5 SOLUTION Solve and check: 1) Simplify the algebraic expressions on each side. Multiply both sides by the LCD: 30 Distributive Property

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Blitzer, Algebra for College Students, 6e – Slide #13 Section 1.4 Solving Linear Equations Cancel CONTINUED Multiply Distribute

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Blitzer, Algebra for College Students, 6e – Slide #14 Section 1.4 Solving Linear EquationsCONTINUED Combine like terms14x + 2 = 15x 2) Collect variable terms on one side and constant terms on the other side. 2 = x Subtract 14x from both sides 14x – 14x + 2 = 15x – 14x Simplify 3) Isolate the variable and solve. Already done.

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Blitzer, Algebra for College Students, 6e – Slide #15 Section 1.4 Solving Linear EquationsCONTINUED 4) Check the proposed solution in the original equation. Replace x with 2 Simplify Original Equation Simplify ? ? ? ?

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Blitzer, Algebra for College Students, 6e – Slide #16 Section 1.4 Solving Linear EquationsCONTINUED 1 - 0 = 1 Simplify 1 = 1 Simplify Since the proposed x value of 2 made a true sentence of 1 = 1 when substituted into the original equation, then 2 is indeed a solution of the original equation.

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Blitzer, Algebra for College Students, 6e – Slide #17 Section 1.4 Solving Linear EquationsEXAMPLE SOLUTION 5 + 4x = 9x + 5 5 - 5 + 4x = 9x + 5 - 5 4x = 9x 4x – 4x = 9x – 4x Subtract 5 from both sides Simplify Subtract 4x from both sides More Practice 0 = 5x Simplify Divide both sides by 5 0 = x

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1.4 Assignment (part 2) p. 46 (26-36 even)

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