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

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

One Step Equations – Multiplication ––––– ––––– –––– – ––– –– –––– ––– 7b = –28 Draw a vertical line and horizontal line To get b by itself. 1. What’s happening to b ? to b ? b times 7. * It’s b times 7. * The opposite of b times 7 is b divided by 7 b divided by 7, so 2.Divide both sides by 7. by 7. 7 –– –– –– –– –– –– –– –– –– –– –– –– –– –– 7 b = –4 b = –4 3. Check Replace b with –4 Do the math Are both sides equal? Rewrite the equation –4 ✓

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

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

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

One Step Equations w/ Fractions – Adding/Subtracting

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

One Step Equations w/ Fractions – Multiplying/Dividing

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

Two–Step Equations – Multiplication +14 = Two–Step Equations – Division – 12 4 – = – 2x –2 x 2= 4 x= 44 2 x = 16 – a6 ‒ 16 ‒ 10 = – a Remember, ‒ a = ‒ 1a So, stick a 1 in front of the a. 1 ‒ 10 = – a 1 ‒1‒1 ‒1‒1 10 = a 9 = ‒ y If you have a negative sign just sitting in front of a fraction, move it next to the constant. 9 = y + 12 ‒ 12 ‒ 12 ‒3‒3 = ‒7 ‒ = y x = ‒ 3 = ‒ 27 + y 8 = y –3 192 –7

EXAMPLE 2 Negative six, increased by the product of four and a number, is negative twenty-two. n = –4 Negative six + the product of four and a number –64n4n= Fifteen is twenty-six less than the quotient of a number and negative three. Writing and Solving a Two-Step Equation increased byisnegative twenty-two. – n = –16 4 = 4 The number is negative four. Fifteen 15 is = twenty-six 26 less than – the quotient of a number and negative three. n – n_ –3 (–3) = = –123n The number is negative one hundred seventeen.

Writing and Solving a Two-Step Equation Your online music website charges a monthly fee of $8, plus $0.35 for every s ss song you download. If you paid $13.25 last month, how many s ss songs did you download? 1. Read it again, and pick out the TOTAL. Set a blank equation equal to = Now, figure out HOW you get to that total. monthly fee + songs = TOTAL x x (songs). 3. Solve for x (songs). Moe, Larry, and Curley are equal partners in a lemonade stand. To calculate each person’s earnings, they’ll take the t tt total money made, divide it by three, then subtract $2 (for supplies). If each stooge got $43, what was the t tt total money made? 1. Read it again, and pick out the TOTAL. Set a blank equation equal to Now, figure out HOW you get to that total. x (total money made). 3. Solve for x (total money made). = 43 total money – supplies = TOTAL 3 x – 2 3 x = 15 You downloaded fifteen songs x = 135 The total money made was $135.

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

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

EXAMPLE 3 6n –2(n +1) = 26 Use Distributive property Combine like terms. 4n = 28 Solve. n = 7 2 Add 2 to each side. 6n –2(n +1) = 26 “outer times first”, “outer times first”, then –2n “outer times second”, –26n6n = 26 4n 4n – 2 = Solving Equations by using Distributive Property

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

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

GUIDED PRACTICE 9x = 12x – 9 2. x = 3 –15x = 15x 3. 4 = x 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)

GUIDED PRACTICE 4. 4a + 5 = a + 11 a = 2 1. Get the variables on one side... …and the constants on the other. …but, which side for each?...it doesn’t really matter. Hint: Move the smaller variable to the larger variable’s side. –a 3a + 5 = + 11 – 5 – 5 5 Subtract 5 to isolate the variable. 3a = 6Solve. 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)

–6c + 1 = –9c c = n = –8 3n + 7 = 2n – x – 7 = 6x + 5 – 3x121. 6x + 5 – 2x = 4 + 4x + 1 there are no solutions for xall values of x are solutions 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)

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