One Step Equations – Addition One Step Equations, p.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 One Step Equations, p.1 Are both sides equal?

One Step Equations – Subtraction One Step Equations, p.1 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 One Step Equations, p.1 Are both sides equal?

One Step Equations w/ Fractions – Adding/Subtracting A. Draw a vertical & horizontal. ●4 4 ●3 9 + a = = B. Covert fractions to a common denominator. ●4 ●3 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)

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

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?

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?

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 x times , ... 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 x = x = –40 3. Multiply the fractions. or

One Step Equations w/ Fractions – Multiplying/Dividing

Two–Step Equations – Multiplication Two–Step Equations – Division 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 – – – – – – – – –

Two–Step Equations – Multiplication Two–Step Equations – Division 6 – 12 ‒16 ‒16 4 – = 2x ‒10 = – a Remember, ‒ a = ‒1a So, stick a 1 in front of the a. 1 –2 –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. +14 9 = y + 12 x 2 = 22 2 –7 2 ‒12 ‒ 12 x = 44 = ‒7 –7 ‒3 21 = y 192 = y

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

– –6 4n = –22 +6 +6 4n = –16 4 = 4 n = –4 n 15 26 –3 + 26 + 26 (–3) 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 =

Writing and Solving a Two–Step Equation Your online music website charges a monthly fee of $8, plus $0.35 for every song you download. If you paid $13.25 last month, how many songs did 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

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

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

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 2 to each side. 4n = 28 Solve. n = 7

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

Solving Equations Using Square Roots The solutions are 2 = x 64 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 2 = x 64 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. = + – 64 Don’t believe me? Check it. = x + – 8 Evaluate square roots. When you find the square root of a fraction, find the square root of each part separately. x2 = ANSWER The solutions are 8 and –8. x2 = x = x =

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

Solving Equations Using Square Roots Solving Equations Square Roots 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 r gives the speed needed to keep riders pinned to the wall. In the model, s is the speed in meters per second and r is 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. r s = 4.95 Write equation for speed of the ride. = 4.95 2.61 Substitute 2.61 for r. 4.95 (1.62) Approximate the square root using a calculator. = 8.019 Multiply. ANSWER The speed should be about 8 meters per second.

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

Solving Equations with Variables on Both Sides* *(not taught in Math 7) GUIDED PRACTICE 55 + 3x = 8x 1. 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

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

...it doesn’t really matter. 3a + 5 = + 11 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) GUIDED PRACTICE 1. Get the variables on one side... 4. 4a + 5 = a + 11 …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 5 to isolate the variable. 3a = 6 Solve. a = 2

Solving Equations with Variables on Both Sides *(not taught in Math 7) Solving Equations with Variables on Both Sides* *(not taught in Math 7) 3n + 7 = 2n –1 118. 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

122. 4(w – 9) = 7w + 18 123. 2(y + 4) = –3y – 7 w = –18 y = –3 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

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

Solving Multi–Step Equations 3n + 7 = 2n –1 118. 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

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

Writing and Solving Multi–Step Equations 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. Let x represent the price of one tube of wax. 39.95 + 4x = 54.95 + 2x Write an equation. 39.95 + 2x = 54.95 Subtract 2x from each side. 2x = 15.00 Subtract 39.95 from each side. 2x 2 15.00 = Divide each side by 2 x = 7.50 Solve. ANSWER The price of one tube of wax is $7.50.

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 both sides = 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?