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**2.1 – Linear Equations in One Variable**

Objective – Solve linear equations using properties of equality. Solve linear equations that can be simplified by combining like terms. -Solve linear equations involving fractions

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Why is this important?

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**Linear Equations Linear Equations in one variable –**

A linear equation in one variable is an equation that can be written in the form ax + b = c Where a, b, and c are real numbers and a 0 The Addition and Multiplication Properties of Equality If a, b, and c, are real numbers, then a=b and a+c = b – c are equivalent equations Also a =b and ac = bc are equivalent equations as long as c

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**Solve: Addition property of equality**

2 x + 5 = 9 2x = 9 -5 2x = 4 2 2 x = 2 Subtract 5 from both sides Simplify Divide both sides by 2 Simplify

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Check To see that 2 is the solution, replace x in the original equation with 2. 2 x + 5 = 9 2 (2) + 5 = 9 4 + 5 = 9 9 = 9

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Give it a try 3 x + 6 = 12 3x + 6 – 6 = 12 –6 3x = 6 3 3 x = 2

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Check 0.6 = 2 – 3.5c 0.6 = 2 – 3.5 (0.4) 0.6 = 2 – 1.4 0.6 = 0.6

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**Give it a try! 4.5 = 3 + 2.5 x 4.5 – 3 = 3 + 2.5 x – 3 1.5 = 2.5 x**

0.6 = x

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**Solve – Combining like terms**

-6 x – 1 + 5x = 3 -6x – 1 + 5x = 3 -x – 1 = 3 -x – = 3 + 1 -x = 4 x = -4

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**Give it a try! -2x + 2 – 4x = 20 -6x + 2 = 20 -6x + 2 – 2 = 20 – 2**

x = -3

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**Solve: Distributive Property**

2 (x – 3) = 5 x – 9 2(x – 3) = 5 x – 9 2x – 6 = 5 x – 9 2x – 6 – 5x = 5x –9 – 5x -3 x – 6 = - 9 -3 x – = -3x = -3 x = 1

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Give it a try! 4 (x – 2) = 6x –10 4x – 8 = 6x – 10 2 = 2 x 1 = x

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**Solve: Adding/Subtracting Fractions**

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Give it a try!

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**Solve: Multiplying Fractions**

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Give it a try!

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Solve: Decimals 0.3x = 0.27 x – 0.02 100(0.3x +0.1) = 100 (0.27x – 0.02) 100(0.3x) (0.1) = 100(0.27x) – 100(0.02) 30 x = 27 x – 2 30 x – 27 x = -2 –10 3 x = -12 3x = -12 x = - 4

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Give it a try! 0.2x +0.1 = 0.12x – 0.06 100 (0.2 x + 0.1) = 100 (0.12 x – 0.06) 20 x + 10 = 12 x – 6 20 x – 12 x = -6 – 10 8 x = -16 x = -2

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**Solve: Contradiction 3x + 5 = 3(x+2) 3 x + 5 = 3 (x + 2)**

5 = 6 This is a false statement…The original equation has no solution. Its solution set is written either by { } or O. This equation is a contradiction.

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Give it a try! 5 x – 1 = 5(x+3) 5 x – 1 = 5 x + 15 -1 = 15 { }

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**This equation is called an identity!**

Solve: Identity 6 x – 4 = (x –1) 6x – 4 = 2 + 6x – 6 6 x – 4 = 6 x – 4 {x/ x is a real number} 6 x – – 6x – 4 + 4 6 x = 6 x 6x – 6x = 6 x – 6x 0 = 0 This equation is called an identity!

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**Give it a try! -4(x – 1) = -4x –9 +13 -4x + 4 = -4x + 4**

{x / x is a real number}

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**Put it in words! Solving a linear equation in one variable**

Step 1: Clear the equation of fractions by multiplying both sides of the equation by the LCD Step 2: Use distributive property to remove grouping symbols such as parentheses. Step 3: Combine like terms. Step 4: Isolate the variable by adding, subtracting, multiplying, dividing (equality properties) Step 5: Check the solution in the original equation.

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