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Published byArline Richardson Modified over 9 years ago
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SOLVING EQUATIONSThe single most important skill in algebraGoal: Find the one value of the variable that makes the sentence true.
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For instance, if 2x + 3 = 17, then 7 is the only value of x that will make this a true sentence. So x = 7.
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We can solve equations by doing the OPPOSITE of what has been done to the variable in the problem.If a problem says +, you subtract.If a problem has multiplication, you divide.
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By doing the opposite, we keep the sides of the equation balanced.
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As long as you do the SAME thing to both sides of an equation, it will remain balanced.
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7x – 13 = 50
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7x – 13 = 50 +13 +13 7x = 63
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7x – 13 = 50 +13 +13 7x = 63 7 7 x = 9
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-5x + 7 = 82
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-5x + 7 = 82 - 7 - 7 -5x = 75
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-5x + 7 = 82 - 7 - 7 -5x = 75 -5 -5 x = -15
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4x + 25 = 13
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4x + 25 = 13 - 25 -25 4x = -12
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4x + 25 = 13 - 25 -25 4x = -12 4 4 x = -3
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When you solve equations, you also do the opposite of the order of operations. Add/subtract first Then divide/multiply
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Solve these equations: 4a + 11 = 59 -2b + 13 = 5 5c – 72 = 98 -4d + 11 = 47
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Solve these equations: 4a + 11 = 59 a = 12 -2b + 13 = 5 b = 4 5c – 72 = 98 c = 34 -4d + 11 = 47 d = -9
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Solve these equations: 5x – 18 = 40 2y + 73 = 54 -3z + 5 = 1
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Solve these equations: 5x – 18 = 40 x = 58 / 5 or 11.6 2y + 73 = 54 y = -19 / 2 or -9.5 _ -3z + 5 = 1 z = 4 / 3 or 1.3
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Some equations are even easier. n + 4 = 13 5x = 35
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Some equations are even easier. n + 4 = 13 Just subtract 4 … x = 9 5x = 35 Just divide by 5 … x = 7
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Solve 17 = 3x – 7
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Solve 17 = 3x – 7 +7 +7 24 = 3x
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Solve 17 = 3x – 7 +7 +7 24 = 3x 3 3 8 = x
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Solve 19 – 2x = 104
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Solve 19 – 2x = 104 -19 -19 -2x = 85
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Solve 19 – 2x = 104 -19 -19 -2x = 85 -2 -2 x = -85 / 2 or -42.5
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If you know the basic steps, you can quickly do equations with more difficult numbers using a calculator.
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12x + 1794 = 2127
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963 – 25x = 704
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What about this?
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Fractions mean division, so to cancel, we’ll subtract 13 and then multiply by 7. n = 63
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Things that can complicate solving equations …
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ParenthesesUse distributive property first. Like termsCombine them first.
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3(2x – 5) = 27
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3(2x – 5) = 27 6x – 15 = 27 6x = 42 x = 7
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-7(2x – 11) = 98
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-7(2x – 11) = 98 -14x + 77 = 98 -14x = 21 x = -3 / 2 or -1.5
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4p + 3 – 2p + 7 + 5p + 2 = 17
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4p + 3 – 2p + 7 + 5p + 2 = 17 7p + 12 = 17 7p = 5 p = 5/ 7
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5(3x + 5) – 3(2x – 1) = 145
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5(3x + 5) – 3(2x – 1) = 145 15x + 25 – 6x + 3 = 145
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5(3x + 5) – 3(2x – 1) = 145 15x + 25 – 6x + 3 = 145 9x + 28 = 145
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5(3x + 5) – 3(2x – 1) = 145 15x + 25 – 6x + 3 = 145 9x + 28 = 145 9x = 117 x = 13
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The goal is always to simplify. Make the problem look like the easy ones we know how to solve.
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Variable on Both SidesFind the smaller number of the variable, and subtract that on both sides.Solve the remaining problem.
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7x – 15 = 2x + 70
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7x – 15 = 2x + 70 -2x -2x 5x – 15 = 70
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7x – 15 = 2x + 70 -2x -2x 5x – 15 = 70 5x = 85 x = 17
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5x + 13 = 7x + 40
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5x + 13 = 7x + 40 -5x -5x 13 = 2x + 40
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5x + 13 = 7x + 40 -5x -5x 13 = 2x + 40 x = -27 / 2 or -13.5
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3(2x + 7) = 3x + 4 + x + 9
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3(2x + 7) = 3x + 4 + x + 9 6x + 21 = 4x + 13
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3(2x + 7) = 3x + 4 + x + 9 6x + 21 = 4x + 13 2x + 21 = 13 2x = -8 x = -4
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Yesterday you saw this equation: 3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x) Solve it.
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3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x) 3x + 20x – 10 – 2x + 8 = 8x – 28 + 12x
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3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x) 3x + 20x – 10 – 2x + 8 = 8x – 28 + 12x 21x – 2 = 20x – 28
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3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x) 3x + 20x – 10 – 2x + 8 = 8x – 28 + 12x 21x – 2 = 20x – 28 -20x -20x x – 2 = -28 +2 +2 x = -26
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Special equations 2(3x – 7) = 6x + 11 10x – 15 = 5(2x – 3)
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2(3x – 7) = 6x + 11 6x – 14 = 6x + 11 ?????
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10x – 15 = 5(2x – 3) 10x – 15 = 10x – 15 ?????
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When variables cancel out…If you have the exact same thing on both sides (like 8 = 8), the answer is ALL REAL NUMBERS or INFINITELY MANY SOLUTIONS. 10x – 15 = 5(2x – 3) 10x – 15 = 10x – 15 -15 = -15
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An equation with infinitely many solutions can also be called an IDENTITY.
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If there is something different on the 2 sides (like 5 = 7), there is NO SOLUTION. 2(3x – 7) = 6x + 11 6x – 14 = 6x + 11 -14 = 11
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