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## Solving Equations

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**Solving Equations**by Lauren McCluskey**Credits**Prentice Hall Algebra I**Solving One-Step Equations:**“An equation is like a balance scale because it shows that two quantities are equal. The scales remained balanced when the same weight is added (or removed from) to each side.” x + 2 = 5 x + (-5) = -2**What does it mean to Solve an equation?**“To solve an equation containing a variable, you find the value (or values) of the variable that make the equation true.” “Get the variable alone on one side of the equal sign…using inverse operations, which are operations that undo each other.”**Inverse Operations**Addition and Subtraction are inverse operations because they undo each other. Multiplication and division are inverse operations because they undo each other.**Properties of Equality:**• Addition Property of Equality: “For every real number a, b, and c, if a = b, then a + c = b + c.” • Subtraction Property of Equality: “For every real number a, b, and c, if a = b, then a - c = b - c.”**Properties of Equality:**• Multiplication Property of Equality: “For every real number a, b, and c, if a = b, then a*c = b*c.” • Division Property of Equality: “For every real number a, b, and c, if a = b, then a/c = b/c.”**Using Reciprocals:**2/3x = 12 In order to solve the equation above, you need to divide by 2/3. Remember: To divide a fraction, you multiply by its reciprocal. In other words: flip it! 3/2*2/3x= 1x 3/2* 12/1= 18 So x = 18.**Examples:**-14 • a - 4 = -18 • b + 24 = 19 • v/ 3 = -4 • 15c = 90 • -1/5 r = -4 -5 -12 6 +20**Try It!**• d + (-4) = -7 • -54 = q - 9 • m / -4 = 13 • -75 = -15x • 2/3 n = 14**Check your answers:**• -3 • -45 • -52 • +7 • +21**Solving Two-Step Equations**“A two-step equation is an equation that involves two operations.” PEMDAS tells us to multiply or divide before we add or subtract, but to solve equations, we do just the opposite: we add or subtract before we multiply or divide.**Try It!**• 7 = 2y - 3 • 6a + 2 = -8 3) x/9 - 15 = 12 4) -x + 7 = 12 5) a -5 = -8 6) 4 = -c + 11**Check your answers:**• 7 = 2y - 3 +3 +3 10 = 2y 2 2 5= y 2) 6a + 2 = - 8 -2 - 2 6a = -10 6 6 a = -1 2/3**To Solve Multi-Step Equations:**“Clear the equation of fractions and decimals.” Apply the Distributive Property as needed. “Combine like terms.” “Undo addition and subtraction.” “Undo multiplication and division.”**Example:**1/5 + 3w/ 15 = 4/5 To clear fractions: * every term by 5 5 * 1/5 = 1; 5* 3w/15 = 9w; 5 * 4/5= 4 So 1 + 9w = 4 Undo addition: -1 -1 9w = 3 Undo multiplication: 9w / 9 = 3 / 9 So w = 1/3**Try It!**• a/7 - 5/7 = 6/7 • 9y/ 14 + 3/7 = 9/14 • 2/3 + 3/k = 71/12**Check your answers:**• a = 11 • y = 1/3 • k = 4/7**Example:**0.11p + 1.5 = 2.49 Clear decimals by * by 100 100 * 0.11= 11; 100 * 1.5 = 150; 100 * 2.49 = 249 So 11p + 150 = 249 Undo addition: -150 -150 11p = 99 Undo multiplication: 11p/11 = 99/11 p = 9**Try it!**• 25.24 = 5y + 3.89 • 0.25m + 0.1m = 9.8 • 26.54 - p = 0.5(50 - p)**Check your answers:**• y = 4.27 • m = 28 • p = 3.08**Try it!**• -4(x + 6) = -40 • m + 5(m -1) = 7 • 1/4(m - 16 ) = 7**Check your answers:**• x = 4 • m = 2 • m = 12**Equations with Variables on Both Sides:**Use the Addition or Subtraction property of Equality to get the variables on one side of the equation.**Example:**4p - 10 = p + 3p -2p Combine like terms: p + 3p - 2p = 2p Use the subtraction property of equality: 4p - 10 = 2p -2p -2p 2p -10 = 0**Example: cont.**2p - 10 = 0 Undo subtraction: +10 +10 2p = 10 Undo multiplication: 2p / 2 = 10/ 2 p = 5**Try It!**• 6b + 14 = -7 - b • -36 + 2w = -8w + w • 30 - 7z = 10z - 4**Check your answers:**• b = -3 • w = 4 • z = 2**Identity or No Solution:**“An equation has no solution if no value of the variable makes the equation true.” “An equation that is true for every value of the variable is an identity.”**2.5: Defining One Variable in Terms of Another:**• “Some problems involve two or more unknown quantities. To solve such problems, first decide which unknown quantity the variable will represent. Then express the other unknown quantity in terms of that variable.”**Example:**“The width of a rectangle is 2 cm less than its length. The perimeter of the rectangle is 16cm. What is the length of the rectangle?” Let l = length Let l - 2 = width P= 2l + 2w So 2(l) + 2(l -2) = 16cm**Example: cont.**2l + 2(l-2) = 16 cm 2l + 2l - 4 = 16cm 4l - 4 = 16 cm +4 +4 4l = 20 cm 4l / 4 = 20/ 4 so l = 5**Try it!**The length of a rectangle is 6 more than 3 times as long as the width. The perimeter is 36 m. What are the measurements?**Check your answer:**l = 15m w = 3m OR: 15m x 3m**Consecutive Integers:**“Consecutive integers differ by 1.” Consecutive even or consecutive odd integers differ by 2. Example: The sum or two consecutive odd integers is 84. What are the integers?**Consecutive Integers**Let x = the 1st integer Let x + 2 = the 2nd integer x + x + 2 = 84 2x + 2 = 84 -2 -2 2x = 82 x = 41; x + 2 = 43 So the integers are 41 and 43.**Try It!**The sum of three consecutive integers is 48. What are the integers?**Check your answer:**The integers are: 15, 16, and 17.**Rate* Times = Distance**• When the distances covered are equal, we can set the two expressions equal to each other and solve for x. • When the distances combine to make up the total distance, we can add the expressions, set it equal to the total distance, and solve for x.**Try It!**Adapted from Prentice Hall: 1) A group of campers left the campsite in a canoe going 10km/h. Two hours later, another group left in a motor boat going 22km/h. How long did it take the second group to catch up?**Try It!**2) On his way to work, your uncle averaged 20 mph. On his way home, he averaged 40mph. If the total time was 1 1/2hours, how long did it take him to drive to work?**Try It!**3) Sarah and John left Perryville going in opposite directions. Sarah drives 12mph faster than John. After 2 hours, they are 176 miles apart. Find Sarah’s and John’s speeds.**Check your answers:**• 1 3/4 hours • 1 hour • John= 33mph Sarah= 45mph