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Transcript
Simplifying with variables Variables cont’d We simplify variables similar to numbers meaning: Ex. 2x + 3x 5x We add the numbers 2 and 3 because they both have an “x” and put the “x” on the end. Examples • 4x + 2x 6x What about this one? 5m + 2m + m – 10m 7m + m – 10m 8m – 10m -2m 12m – 17m -5m 2 variables Ex. 2s + 3r – 4s + 6r With these we arrange the like variables together and simplify ***Make sure to keep the sign in front of them**** 2s – 4s + 3r + 6r -2s + 9r Do we go any further? NO You do • 5x – 13x + 2x 6x – 4y + x – 7w • 10s – 20r + 4r – 6s 14z + 5y - 9z+10y Mathematical Properties • Associative Property – – a + (b + c) = (a + b) + c or (ab)c – a(bc) • Commutative – s+7+r=7+s+r or t*r*s = s*t*r • Identity – 16 + 0 = 16 or r * 1 = r Mathematical Prop. Con’t • Substitution– Ex. 6+ 4 = 10 or 8 – 5 = 3 or 7(3) = 21 • Distributive– a(b + c) = ab * ac – s(r – t + v) = sr – st + sv Prerequisite Skills SKILL CHECK Solve the equation using mental math. 5. x –1 = 5 ANSWER 6 6. x –2 = 9 ANSWER 11 7. 4 + x = 12 ANSWER 8 8. 10 = x + 3 ANSWER 7 9. 5x = 20 ANSWER 4 Prerequisite Skills SKILL CHECK 10. x 3=1 ANSWER 3 11. x 7=5 ANSWER 35 12. 3x = 51 ANSWER 17 Solving equations • Remember “PEMDAS” in simplifying we do the opposite “SADMEP” when solving. • ***Following this process you will never get a problem wrong.*** SADMEP • SADMEP means do the opposite of what you see in that order, to the variable. • Subtraction – going to add • Addition – going to subtract • Division – going to multiply • Multiplication – going to divide • Exponents – going to take the root • Parentheses – do what’s inside last Solving an Equation Using Subtraction EXAMPLE 1 x+ 8 = –15 x+ 8 = –15 –8 x –8 = –23 ANSWER The solution is –23. Original equation Subtract 8 from each side to undo addition. Simplify. x is by itself. EXAMPLE 2 Solving an Equation Using Addition c – 4.5 = 13 c – 4.5 = 13 + 4.5 = + 4.5 c = 17.5 Check 17.5 – 4.5 =? 13 13 = 13 ✓ Add 4.5 to each side to undo subtraction. Simplify. c is by itself. Substitute 17.5 for c in original equation. EXAMPLE 3 Using a Model Rock Climbing A cliff has a height of about 1500 feet. If you have already climbed 675 feet, how much farther do you have to climb to reach the top? SOLUTION Use the diagram to help you write an algebraic model. Let x represent the distance left to climb. EXAMPLE 3 Using a Model 1500 = x + 675 1500 – 675 = x + 675 – 675 825 = x ANSWER You have about 825 feet left to climb. Write an algebraic model. Subtract 675 from each side. Simplify. x is by itself. for Examples 1, 2, and 3 GUIDED PRACTICE Solve the equation. Check your solution. 1. x + 9 = 20 x + 9 = 20 c + 9 – 9 = 20 – 9 x = 11 Check 11 + 9 =? 13 11 = 11 ✓ Original equation Subtract 9 from each side. Simplify. x is by itself. Substitute 11 for x in original equation. for Examples 1, 2, and 3 GUIDED PRACTICE 2. –10 = 3 + y –10 = 3 + y –3 –3 Original equation Subtract 3 from each side. –13 = y Check Simplify. y is by itself. Substitute –13 for y –10 =? 3 – 13 –10 = –10 ✓ in original equation. for Examples 1, 2, and 3 GUIDED PRACTICE 3. m – 14 = –15 m – 14 = –15 + 14 +14 m = –1 Check –1 – 14 =? –15 –1 = – 1 ✓ Original equation Add 14 from each side. Simplify. m is by itself. Substitute –1 for m in original equation. for Examples 1, 2, and 3 GUIDED PRACTICE 4. 2 = z – 6.4 2 = z – 6.4 2 + 6.4 = z – 6.4 + 6.4 8.4 = z Check Original equation Subtract 6.4 from each side. Simplify. z is by itself. 2 =? 8.4 – 6.4 Substitute 8.4 for z 2=2✓ in original equation. GUIDED PRACTICE 5. for Examples 1, 2, and 3 Seashells Lucinda combines her 49 seashells with Jerry’s seashells, for a total of 162. Write and solve an addition equation to find how many seashells Jerry had before their collections were combined. SOLUTION Let s represent Jerry’s seashells s + 49 = 162 s + 49 – 49 = 162 – 49 s = 113 ANSWER Write an algebraic model. Subtract 49 from each side. Simplify. s is by itself. Jerry had 113 seashells before their collections were combined Examples for Multiplying 2x = 8 **Remember a letter and number together is multiplication. 2x = 8 2x = 8 2 2 x=4 We see multiply so divide both sides by 2 The 2’s cancel and 8 divided by 2 is 4 More examples 3x = 18 3x = 18 3 3 x = 6 -5x = 20 -5x = 20 -5 -5 x = -4 • • • • 4x = -12 10x = 90 -8x = 48 -3x = -21 Examples Dividing • We keep our same rules of SADMEP • So b/c we see divide we multiply both sides as the review we did earlier. y 3 =5 1. 119 = 31 + h 2. 59 = 65 - a 3. e ÷ 3 = 2 • 4. 9g = 27 5. u - 11 = 9 6. 21 + w = 30 7. 9 = x ÷ 8 8. 62 = z + 54 9. m + 25 = 115 10. 36 = j - 32 11. 62 = 93 - d 12. 36 ÷ f = 6 13. 139 = b + 98 14. q + 71 = 136 •