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Solving Linear Equations

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To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5 is solved for x. y = 2x - 1 is solved for y.

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Solving Equations Using Addition and Subtraction

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Addition Property of Equality For any numbers a, b, and c, if a = b, then a + c = b + c. What it means: You can add any number to BOTH sides of an equation and the equation will still hold true.

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An easy example: We all know that 7 =7. Does 7 + 4 = 7? NO! But 7 + 4 = 7 + 4. The equation is still true if we add 4 to both sides.

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Let’s try another example! x - 6 = 10 Add 6 to each side. x - 6 = 10 +6 +6 x = 16 Always check your solution!! The original problem is x - 6 = 10. Using the solution x=16, Does 16 - 6 = 10? YES! 10 = 10 and our solution is correct.

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What if we see y + (-4) = 9? Recall that y + (-4) = 9 is the same as y - 4 = 9. Now we can use the addition property. y - 4 = 9 +4 +4 y = 13 Check your solution! Does 13 - 4 = 9? YES! 9=9 and our solution is correct.

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How about -16 + z = 7? Remember to always use the sign in front of the number. Because 16 is negative, we need to add 16 to both sides. -16 + z = 7 +16 +16 z = 23 Check you solution! Does -16 + 23 = 7? YES! 7 = 7 and our solution is correct.

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A trick question... -n - 10 = 5 +10 +10 -n = 15 Do we want -n? NO, we want positive n. If the opposite of n is positive 15, then n must be negative 15. Solution: n = -15 Check your solution! Does -(-15)-10=5? Remember, two negatives = a positive 15 - 10 = 5 so our solution is correct.

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Subtraction Property of Equality For any numbers a, b, and c, if a = b, then a - c = b - c. What it means: You can subtract any number from BOTH sides of an equation and the equation will still hold true.

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3 Examples: 1) x + 3 = 17 -3 -3 x = 14 Does 14 + 3 = 17? 2) 13 + y = 20 -13 -13 y = 7 Does 13 + 7 = 20? 3) z - (-5) = -13 Change this equation. z + 5 = -13 -5 -5 z = -18 Does -18 -(-5) = -13? -18 + 5 = -13 -13 = -13 YES !

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Try these on your own...

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The answers...

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Solving Equations Using Multiplication and Division

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An easy example: We all know that 3 = 3. Does 3 4 = 3? NO! But 3 4 = 3 4. The equation is still true if we multiply both sides by 4.

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Let’s try another example! x = 4 2 Multiply each side by 2. 2 x = 4 2 2 x = 8 Always check your solution!! The original problem is x = 4 2 Using the solution x = 8, Is x/2 = 4? YES! 4 = 4 and our solution is correct.

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A fraction times a variable: The two step method: Ex: 2x = 4 3 1. Multiply by 3. (3)2x = 4(3) 3 2x = 12 2. Divide by 2. 2x = 12 2 2 x = 6 The one step method: Ex: 2x = 4 3 1. Multiply by the RECIPROCAL. (3)2x = 4(3) (2) 3 (2) x = 6

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The two negatives will cancel each other out. The two fives will cancel each other out. (-5) x = -15 Does -(-15)/5 = 3? What do we do with negative fractions? Recall that Solve. Multiply both sides by -5.

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Try these on your own...

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Division Property of Equality 4 For any numbers a, b, and c (c ≠ 0), if a = b, then a/c = b/c What it means: 4 You can divide BOTH sides of an equation by any number - except zero- and the equation will still hold true.

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2 Examples: 1) 4x = 24 Divide both sides by 4. 4x = 24 4 4 x = 6 Does 4(6) = 24? YES! 2) -6x = 18 Divide both sides by -6. -6y = 18 -6 -6 y = -3 Does -6(-3) = 18? YES!

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The answers...

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Solving Equations with the Variable on Both Sides

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To solve these equations, Use the addition or subtraction property to move all variables to one side of the equal sign. Solve the equation using the methods we mentioned.

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Let’s see a few examples: 1) 6x - 3 = 2x + 13 -2x -2x 4x - 3 = 13 +3 +3 4x = 16 4 4 x = 4 Be sure to check your answer! 6(4) - 3 =? 2(4) + 13 24 - 3 =? 8 + 13 21 = 21

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Let’s try another! 2) 3n + 1 = 7n - 5 -3n -3n 1 = 4n - 5 +5 +5 6 = 4n 4 4 Reduce! 3 = n 2 Check: 3(1.5) + 1 =? 7(1.5) - 5 4.5 + 1 =? 10.5 - 5 5.5 = 5.5

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Here’s a tricky one! 3) 5 + 2(y + 4) = 5(y - 3) + 10 Distribute first. 5 + 2y + 8 = 5y - 15 + 10 Next, combine like terms. 2y + 13 = 5y - 5 Now solve. (Subtract 2y.) 13 = 3y - 5 (Add 5.) 18 = 3y (Divide by 3.) 6 = y Check: 5 + 2(6 + 4) =? 5(6 - 3) + 10 5 + 2(10) =? 5(3) + 10 5 + 20 =? 15 + 10 25 = 25

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Let’s try one with fractions! 4) 3 - 2x = 4x - 6 3 = 6x - 6 9 = 6x so x = 3/2 Steps: Multiply each term by the least common denominator (8) to eliminate fractions. Solve for x. Add 2x. Add 6. Divide by 6.

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Two special cases: 6(4 + y) - 3 = 4(y - 3) + 2y 24 + 6y - 3 = 4y - 12 + 2y 21 + 6y = 6y - 12 - 6y - 6y 21 = -12 Never true! 21 ≠ -12 NO SOLUTION! 3(a + 1) - 5 = 3a - 2 3a + 3 - 5 = 3a - 2 3a - 2 = 3a - 2 -3a -2 = -2 Always true! We write IDENTITY.

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Try a few on your own: 9x + 7 = 3x - 5 8 - 2(y + 1) = -3y + 1 8 - 1 z = 1 z - 7 2 4

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The answers: x = -2 y = -5 z = 20

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