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One Step Equations – Addition

One Step Equations – Addition. Draw a vertical line. –. –. –. –. –. x. + 8. = –15. +. +. +. +. –. –. –. –. –. and horizontal line. +. +. +. +. –. –. –. –. –. To get x by itself. +. +. 1. Get rid of + 8. –. –. –. –. – 8. – 8. –. –. –. –. –. –. –. –.

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One Step Equations – Addition

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  1. One Step Equations – Addition Draw a vertical line – – – – – x + 8 = –15 + + + + – – – – – and horizontal line + + + + – – – – – To get x by itself. + + 1. Get rid of + 8 – – – – –8 –8 – – – – – – – – – – – – • How? Add the • opposite • but, what you do • to one side ... • ... you’ve got to • do to the other – – – – – x = –23 – – – – – – – – – – 2. Cancel opposites. – – – – – – – – 3. Add 4. Check – – – • Rewrite the equation – – – – – – – – – – ✓ • Replace x with –23 – – – – – – – – – – • Do the math • Are both sides equal?

  2. One Step Equations – Subtraction Draw a vertical line – x – 7 = –2 – – – and horizontal line – – – – – To get x by itself. 1. Get rid of – 7(or –7) + +7 +7 + + + + + + + • How? Add the • opposite + + + + + + • but, what you do • to one side ... • ... you’ve got to • do to the other x = +5 2. Cancel opposites. + + + + + 3. Add 4. Check • Rewrite the equation ✓ • Replace x with 5 • Do the math • Are both sides equal?

  3. One Step Equations w/ Fractions – Adding/Subtracting A. Draw a vertical & horizontal. 4 9 ●3 ●4 + a = B. Covert fractions to a common denominator. = ●3 ●4 The Right Way: 1. List multiples of both denominators (bottom) * 6: 6, 12, 18, 24, 30, 36, 42, 48 ... * 8: 8, 16, 24, 32, 40, 48, ... 2. The smallest number in both lists is .. a = 3. ...so, that’s your new denominator (bottom). 4. To find your new numerators (tops): Check: = • Whatever you multiplied to get the new • denominator (bottom)... ✓ = • ... multiply the numerator (top) by the • same thing. C. Isolate a. Get rid of . The Lazier Way: • Multiply the denominators (bottoms). • * That’s your new denominator (bottom). D. Add its opposite to both sides. 2. Go to Step 4 to find the new numerators (tops)

  4. One Step Equations w/ Fractions – Adding/Subtracting 1. 2. 3. 4. 5. 6. 7. 8.

  5. One Step Equations – Multiplication Draw a vertical line and horizontal line – – – 7b = –28 – – – – – – – – – – – – – – – – To get b by itself. – – – – – – – – 1. What’s happening to b ? 7 7 – – – – * It’s b times 7. – – – – – – – – – – * The opposite of b times 7 is b divided by 7 , so – – – – – – – – b = –4 – – – – – – • Divide both sides • by 7. 3. Check • Rewrite the equation •–4 ✓ • Replace b with –4 • Do the math • Are both sides equal?

  6. One Step Equations – Division Draw a vertical line and horizontal line To get a by itself. – – – – – – – – – 1. What’s happening to a ? * It’s divided by 3. 3 • = –9 • 3 * The opposite of a divided by 3 is multiplied by 3, so 2. Multiply both sides by 3. a = –27 ✓ 3. Check ? –27 • Rewrite the equation • Replace a with –27 • Do the math • Are both sides equal?

  7. One Step Equations w/ Fractions – Multiplying/Dividing Draw a vertical & horizontal To get x by itself. * Look at x. What’s happening to it? 30 = * It’s x times ... so to get rid of xtimes , ... 2 1. You have to MULTIPLY by the RECIPROCAL x = x = 15 or A reciprocal is a flipped fraction Check • Rewrite the equation • Replace x with 15 ... and, the reciprocal of + is + • Do the math ✓ • Are both sides equal? 30 3 ... so, MULTIPLY both sides by or 10 40 = 2. Cancel the opposites. 1 3. Multiply the fractions. x = x = –40 or

  8. One Step Equations w/ Fractions – Multiplying/Dividing

  9. Two–Step Equations – Multiplication 3x – 7 =‒1 • Look at the variable side, find the constant, • and get rid of it first. + + + + + + + a constant is a number without a variable – it’s the “naked number” – – – + + + – – – – + 7 +7 + + + + – 2. To get rid of ‒7, add the opposite(+7) 3x = 6 3. Cancel the opposites... + + + + + + … bring down the variable term …then add. 3 3 4. To get rid of the coefficient, 3 …… x = 2 + a coefficient is the number in front of the variable + … DIVIDE both sides by 3 Two–Step Equations – Division 8 + x = ‒10 • Look at the variable side, find the constant, and get rid of it first. – – + + – – – – – – 2 + + – – – – – – + + – – – – – – ‒8 ‒8 + + – – – – – – 2. To get rid of 8, add the opposite(‒8) – – – – x = 2 2 ‒18 2 – – – – – – – – 3. Cancel the opposites... – – – – – – – – – – – – – – – – … drop the variable term …then add. – – – – – – – – x = ‒ 36 – – – – – – – – – 4. To get rid of x divided by 2, … – – – – – – – – – – – – – – – – – – … MULTIPLY both sides by 2 – – – – – – – – –

  10. Two–Step Equations – Multiplication 6 = 16 – a –2 +14 – 12 +14 – 12 – – 4 2x = ‒16 ‒16 ‒10 = – a Remember, ‒ a = ‒1a So, stick a 1 in front of the a. 1 –2 x ‒10 = – a 1 2 = ‒1 ‒1 4 x = –3 10 = a Two–Step Equations – Division 9 = ‒y + 12 7 ‒ 3 = ‒27 + y 8 If you have a negative sign just sitting in front of a fraction, move it next to the constant. 9 = y + 12 x 2 2 –7 = 22 2 ‒12 ‒ 12 x = 44 = ‒7 –7 ‒3 21 = y 192 = y

  11. Two–Step Equations with Fractional Coefficients 5 7 Step 1: Get rid of the CONSTANT on the variable side –11 = 4 – n 5 7 4 + n = –11 – 4 – 4 – 4 – 4 • SUBTRACT 4 • from both sides. Step 2: Get rid of the COEFFICIENTon the variable side ) ( ) ) ( 5 7 ( 7 5 n 7 5 5 7 ( ) 7 5 n = –15 –15 = – 7 5 – – 1 • MULTIPLY BY THE • RECIPROCAL, 7 5 – 105 5 –105 5 n = = n Step 3: Cancel opposites, multiply, then simplify. or or –21 n = = n 21

  12. Writing and Solving a Two–Step Equation EXAMPLE 2 1. Negative six, increased by the product of four and a number, is negative twenty–two. Negative six increased by the product of four and a number is negative twenty–two. –6 4n = –22 + +6 +6 4n=–16 The number is negative four. 4 = 4 n= –4 2. Fifteen is twenty–six less than the quotient of a number and negative three. Fifteen is twenty–six less than the quotient of a number and negative three. n – 15 26 = The number is negative one hundred seventeen. –3 + 26 + 26 (–3) 41 n_ (–3) = –3 –123 n =

  13. Writing and Solving a Two–Step Equation Your online music website charges a monthly fee of $8, plus $0.35 for every songyou download. If you paid $13.25 last month, how many songsdid you download? 1. Read it again, and pick out the TOTAL. monthly fee + songs = TOTAL Set a blank equation equal to 13.25 8 + 0.35x = 13.25 2. Now, figure out HOW you get to that total. You downloaded fifteen songs 3. Solve for x (songs). x = 15 Moe, Larry, and Curley are equal partners in a lemonade stand. To calculate each person’s earnings, they’ll take the total money made, divide it by three, then subtract $2 (for supplies). If each stooge got $43, what was the total money made? total money – supplies = TOTAL 3 1. Read it again, and pick out the TOTAL. Set a blank equation equal to 43 = 43 x– 2 3 2. Now, figure out HOW you get to that total. The total money made was $135. 3. Solve for x (total money made). x = 135

  14. Solving Equations by Combining Like Terms 3x +12 – 4x = 20 Look: There are 2 variable terms … … so, COMBINE LIKE TERMS first. Remember, ‒1x = ‒x but, just leave the 1 there. –1x +12= 20 – 12 – 12 • Look at the variable side, find the constant, • and get rid of it first. –1x = 8 2. To get rid of +12, add the opposite(‒12) –1 –1 3. Cancel the opposites … … bring down the variable term …then add. 4. To get rid of the coefficient, ‒1 … … x = –8 … DIVIDE both sides by ‒1

  15. Solving Equations by Combining Like Terms 3. 4p +10 + p = 25 –8r – 2 + 7r = – 9 2. –6 = 11w –5w 1. Solve the equation. w = – 1 p = 3 r = 7

  16. Solving Equations by using Distributive Property EXAMPLE 3 6n –2(n +1) = 26 Use Distributive property “outer times first”, then 6n –2(n +1) = 26 “outer times second”, Combine like terms. 6n –2n –2 = 26 4n – 2 = 26 + 2 + 2 Add 2to each side. 4n = 28 Solve. n = 7

  17. Solving Equations by using Distributive Property 1. 2. 3. 3(x – 9) = – 39 –63 = –7(8 – p) 25 = –3(2x + 1) x = or – 4 x = – 4 p = –1

  18. Solving Equations Using Square Roots x 64 64 = 2 + + = – – x 8 = x 64 2 = The solutions are 8 and –8. ANSWER c2 = 0.0121 When you find the square root of a decimal number, pretend there is no decimal place. Take the square root of both sides. c2 = 0.0121 The square root of 121 is 11, so the square root of 0.0121 must be .11 c = 0.11 Remember, real numbers have 2 roots. Don’t believe me? Check it. Evaluate square roots. When you find the square root of a fraction, find the square root of each part separately. x2 = x2 = x = x =

  19. Solving Equations Square Roots Solving Equations Using Square Roots k 121 t2 36 = 34. 33. 2 = + + + – – – ANSWER ANSWER ANSWER ANSWER x x x 0.09 14 15 11 = = = t = 6 _ + GUIDED PRACTICE

  20. Solving Equations Square Roots Solving Equations Using Square Roots s 4.95 = r 4.95 = 4.95 (1.62) 8.019 2.61 = EXAMPLE 4 37. On an amusement park ride, riders stand against a circular wall that spins. At a certain speed, the floor drops out and the force of the rotation keeps the riders pinned to the wall. The model s = 4.95 rgives the speed needed to keep riders pinned to the wall. In the model, sis the speed in meters per second and ris the radius of the ride in meters. Find the speed necessary to keep riders pinned to the wall of a ride that has a radius of 2.61 meters. Write equation for speed of the ride. Substitute 2.61 for r. Approximate the square root using a calculator. Multiply. The speed should be about 8 meters per second. ANSWER

  21. Solving Equations Square Roots ( )2( )2 ( )2 ( )2 ( )2( )2 ( )2 ( )2 b = 64 144 = a x = 2.89 = y

  22. Solving Equations with Variables on Both Sides* *(not taught in Math 7) 55 + 3x = 8x 1. GUIDED PRACTICE What’s the goal? Get the variables on one side... – 3x – 3x …and the constants on the other. …so, if you get rid of 3x on the left, you’ll have it. 55 = 5x 11 = x Solve. or x = 11

  23. Solving Equations with Variables on Both Sides Solving Equations with Variables on Both Sides* *(not taught in Math 7) Solving Equations with Variables on Both Sides* *(not taught in Math 7) 9x = 12x – 9 2. 3. –15x + 120 = 15x GUIDED PRACTICE x = 3 4 = x

  24. Solving Equations with Variables on Both Sides* *(not taught in Math 7) Solving Equations with Variables on Both Sides Solving Equations with Variables on Both Sides* *(not taught in Math 7) 4. 4a + 5 = a + 11 GUIDED PRACTICE 1. Get the variables on one side... …and the constants on the other. …but, which side for each? –a –a ...it doesn’t really matter. 3a + 5 = + 11 Hint: Move the smaller variable to the larger variable’s side. – 5 – 5 Subtract 5to isolate the variable. 3a = 6 Solve. a = 2

  25. Solving Equations with Variables on Both Sides Solving Equations with Variables on Both Sides* *(not taught in Math 7) Solving Equations with Variables on Both Sides* *(not taught in Math 7) 118. 3n + 7 = 2n –1 119. –6c + 1 = –9c + 7 n = –8 c = 2 120. 11 + 3x – 7 = 6x + 5 – 3x 121. 6x + 5 – 2x = 4 + 4x + 1 there are no solutions for x all values of x are solutions

  26. Solving Equations with Variables on Both Sides Solving Equations with Variables on Both Sides* *(not taught in Math 7) GUIDED PRACTICE 122. 4(w – 9) = 7w + 18 123. 2(y + 4) = –3y – 7 w = –18 y = –3

  27. Solving Multi–Step Equations 117. 4a + 5 = a + 11 GUIDED PRACTICE Get the variables on one side... …and the constants on the other. …but, which side for each? It doesn’t really matter. –a –a Hint: Move the smaller variable to the larger variable’s side. 3a + 5 = + 11 – 5 – 5 Subtract 5to isolate the variable. 3a = 6 Solve. a = 2

  28. Solving Multi–Step Equations 118. 3n + 7 = 2n –1 119. –6c + 1 = –9c + 7 n = –8 c = 2 120. 11 + 3x – 7 = 6x + 5 – 3x 121. 6x + 5 – 2x = 4 + 4x + 1 there are no solutions for x all values of x are solutions

  29. Solving Multi–Step Equations GUIDED PRACTICE 122. 4(w – 9) = 7w + 18 123. 2(y + 4) = –3y – 7 y = –3 w = –18

  30. Writing and Solving Multi–Step Equations Let xrepresent the price of one tube of wax. 2x 15.00 = ANSWER 2 2 The price of one tube of wax is $7.50. 124. You and a friend are buying snowboarding gear. You buy a pair of goggles that costs $39.95 and 4 tubes of wax. Your friend buys a helmet that costs $54.95 and 2 tubes of wax. You each spend the same amount. Write and solve an equation to find the price of one tube of wax. 39.95 + 4x= 54.95 + 2x Write an equation. 39.95 + 2x= 54.95 Subtract 2xfrom each side. 2x = 15.00 Subtract 39.95 from each side. Divide each side by 2 x = 7.50 Solve.

  31. One Step Equations w/ Fractions – Adding/Subtracting Draw a vertical & horizontal + a = = To get x by itself. 1. Get rid of + • How? Add the OPPOSITE • to bothsides = 2. Cancel opposites. 3. Add • NOTE: With fractions, • you must find a • common • denominator . a = Check • Rewrite the equation • Replace a with ✓ • Do the math • Are both sides equal?

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