Translating and Solving Equations

-1 = -6 - 3x add 6 to both sides 5 = -3x divide both sides by -3 1 2/3 = x

-1 = -6 - 3(1 2/3) -1 = -6 - 5 -1 = -1

-x + 8 - x = 3x + 10 - 3 combine like terms -2x + 8 = 3x + 7 subtract 7 from both sides -2x + 1 = 3x add 2x to both sides 1 = 5x divide both sides by 5 0.2 = x

-1(0.2) + 8 - 1x(0.2) = 3(0.2) + 10 - 3 -0.2 + 8 -0.2 = 0.6 + 10 - 3 7.6 = 7.6

-3 (x + 5) + 5 = 5 (x + 4) + 7 -3x - 15 + 5 = 5x + 20 + 7 combine like terms -3x - 10 - 5x + 27 add 10 to both sides -3x = 5x + 37 subtract 5x from both sides -8x = 37 divide both sides by -8 x = -4.625

-3 (x + 5) + 5 = 5 (x + 4) + 7 -3 (-4.625 + 5) + 5 = 5 (-4.625 + 4) + 7 13.875 -15 + 5 = -23.125 + 20 + 7 -1.125 + 5 = -3.125 + 7 3.875 = 3.875

x/7 = -6 multiply both sides by 7 x = -42

-42/7 = -6 -6 = -6

-8x = 0 divide both sides by -8 x = 0

-8(0) = 0 0 = 0

1 - 2x = -5x + 13 subtract 1 from both sides -2x = -5x + 12 add 5x to both sides 3x = 12 divide both sides by 3 x = 4

1 - 2(4) = -5(4) + 13 1 - 8 = -20 + 13 -7 = -7

Solve. 1 = x - 8 add 8 to both sides 9 = x

1 = 9 - 8 1 = 1

1/5x + 8 = 12 subtract 8 from both sides 1/5x = 4 multiply both sides by 5/1 x = 20

1/5(20) + 8 = 12 4 + 8 = 12 12 = 12

1/8x + 3 = -1 subtract 3 from both sides 1/8x = -4 multiply both sides by 8/1 x = -32

1/8(-32) + 3 = -1 -4 + 3 = -1 -1 = -1

10x = -30 divide both sides by 10 x = -3

10(-3) = -30 -30 = -30

10x = -17 + 37 combine like terms 10x = 20 divide both sides by 10 x = 2

10(2) = -17 + 37 20 = 20

Solve. 18 - 8 + x = 8 + 5 - 3 combine like terms 10 + x = 13 - 3 10 + x = 10 subtract 10 from both sides x = 0

18 - 8 + 0 = 8 + 5 - 3 10 = 13 - 3 10 = 10

2 (x + 5) = -12 2(x) + 2(5) = -12 2x + 10 = -12 subtract 10 from both sides 2x = -22 divide both sides by 2 x = -11

2 (-11 + 5) = -12 2(-11) + 2(5) = -12 -22 + 10 = -12 -12 = -12

2/3 = 1/3 + 3x subtract 1/3 from both sides 1/3 = 3x divide both sides by 3 1/9 = x

2/3 = 1/3 + 3(1/9) 2/3 = 1/3 + 1/3 2/3 = 2/3

Solve. 3x - 2 = -23 add 2 to both sides 3x = -21 divide both sides by 3 x = -7

3(-7) - 2 = -23 -21 - 2 = -23 -23 = -23

Thirty is five less than triple a number.

30 = 3n - 5

Triple a number is equal to eight more than five times the number.

3n = 8 + 5n

4 (x + 7) = 36 (4)(1x) + (4)(7) = 36 4x + 28 = 36 subtract 28 from both sides 4x = 8 divide both sides by 4 x = 2

4 (2 + 7) = 36 (4)(2) + (4)(7) = 36 8 + 28 = 36 36 = 36

4 (x + 1) = 28 4(1x) + 4(1) = 28 4x + 4 = 28 subtract 4 from both sides 4x = 24 divide both sides by 4 x = 6

4 (6 + 1) = 28 4(6) + 4(1) = 28 24 + 4 = 28 28 = 28

x/2 + 6 = 8 subtract 6 from both sides x/2 = 2 multiply both sides by 2/1 x = 4

4/2 + 6 = 8 2 + 6 = 8 8 = 8

5x - 18x = 26 combine like terms -13x = 26 divide both sides by -13 x = -2

5(-2) - 18(-2) = 26 -10 + 36 = 26 26 = 26

5x + 3 = 18 subtract 3 from both sides 5x = 15 divide both sides by 5 x = 3

5(3) + 3 = 18 15 + 3 = 18 18 = 18

5x - 4 = 13 add 4 to both sides 5x = 17 divide both sides by 5 x = 3.4

5(3.4) - 4 = 13 17 - 4 = 13 13 = 13

6x - 8 = -2 add 8 to both sides 6x = 6 divide both sides by 6 x = 1

6(1) - 8 = -2 6 - 8 = -2 -2 = -2

6x - 7 = 23 add 7 to both sides 6x = 30 divide both sides by 6 x = 5

6(5) - 7 = 23 30 - 7 = 23 23 = 23

7x = 14 divide both sides by 7 x = 2

7(2) = 14 14 = 14

8 - 7x - 2 = -4 + 5x - 14 combine like terms -7x + 6 = -18 + 5x subtract 6 from both sides -7x = -24 + 5x subtract 5x from both sides -12x = -24 divide both sides by -12 x = 2

8 - 7(2) - 2 = -4 + 5(2) - 14 8 - 14 - 2 = -4 + 10 - 14 -8 = -8

9x + 4 = 7x - 2 subtract 7x from both sides 2x + 4 = -2 subtract 4 from both sides 2x = -6 divide both sides by 2 x = -3

9(-3) + 4 = 7(-3) - 2 -27 + 4 = -21 - 2 -23 = -23

Is -1 a solution for the equation 2x + 6 = -1 ? 2(-1) + 6 = -1 -2 + 6 = 4

No

Is -8 a solution of x + 12 = 20 ?

No -8 + 12 = 4 4 ≠ 20

The sum of a number and fifty is one hundred eighty-eight.

n + 50 = 188

The difference between a number and two is twenty-one.

n - 2 = 21

Thirteen minus four times a number is thirteen.

13 - 4n = 13

13 = 5x - 22 add 22 to both sides 35 = 5x divide both sides by 5 7 = x

13 = 5(7) - 22 13 = 35 - 22 13 = 13

15x - 7 = 2 + 12x subtract 2 from both sides 15x - 9 = 12x subtract 15x from both sides -9 = -3x divide both sides by -3 3 = x

15(3) - 7 = 2 + 12(3) 45 - 7 = 2 + 36 38 = 38

Sixteen minus seven-ninths of a number is twelve.

16 - 7/9n = 12

8x - 4 = -36 add 4 to both sides 8x = -32 divide both sides by 8 x = -4

8(-4) - 4 = -36 -32 - 7 = -36 -36 = -36

9x - 8x - 7 = 3 + x - 10 combine like terms 1x - 7 = x - 7 subtract 1x from both sides -7 = -7

9(-7) - 8(-7) - 7 = 3 + (-7) - 10 -63 + 56 - 7 = 3 + (-7) - 10 -7 - 7 = -4 - 10 -14 = -14

Which step shows how to solve 1/3x = 9? a. Multiply each side of the equation by 3 b. Subtract 1/3 from each side of the equation c. Divide each side of the equation by 9 d. Multiply each side of the equation by 1/3

Multiply each side of the equation by 3

Variable

is a letter or symbol that represents an unknown quantity.

The quotient of a number and negative eight, increased by three, is twenty-seven.

x / (-8) + 3 = 27

In the list of numbers, find the one that is a solution of the given equation. -4, 14, 1 x - 9 = 5

x = 14

9x = 3 (8x + 35) 9x = (3)(8x) + (3)(35) 9x = 24x + 105 subtract 24x from both sides -15x = 105 divide both sides by -15 x = -7

(9)(-7) = 3 ((8)(7) + 35) -63 = 3 (-56 + 35) -63 = -168 + 105 -63 = -63

-14 = -x divide both sides by -1 14 = x

-14 = (14)(-1x) -14 = -14

-15x = -195 divide both sides by -15 x = 13

-15(13) = -195 -195 = -195

In the list of numbers, find the one that is a solution of the given equation. 10, -16, 23 -3 = x + 13

-16

-28x - 13 = 47 add 13 to both sides -28x = 60 divide both sides by -28 x = -2 1/7

-28(-2 1/7) - 13 = 47 60 - 13 = 47 47 = 47

11 (x - 6) = -3 - 8 (11)(1x) + (11)(-6) = -3 - 8 11x - 66 = -11 add 66 to both sides 11x = 55 divide both sides by 11 x = 5

11 (5 - 6) = -3 - 8 55 - 66 = -3 - 8 -11 = -11

11 - 26 = -5x combine like terms -15 = -5x divide both sides by -5 3 = x

11 - 26 = -5(3) -15 = -15

16 - 2x = -4x add 2x to each side 16 = -2x divide both sides by -2 -8 = x

16 + 2(-8) = -4(-8) 16 + 16 = 32 32 = 32

4x + 6 = -6 + 2x + 18 combine like terms 4x + 6 = 12 + 2x subtract 6 for both sides 4x = 6 + 2x subtract 2x from both sides 2x = 6 divide both sides by 2 x = 3

4(3) + 6 = -6 + 2(3) + 18 12 + 6 = -6 + 6 + 18 18 = 18

4x - 39 = 3 - 3x subtract 3 from both sides 4x - 42 = -3x subtract 4x from both sides -42 = -7x divide both sides by -7 6 = x

4(6) - 39 = 3 - 3(6) 24 - 39 = 3 - 18 -15 = -15

5x + 26 - 6 = 9x + 12x combine like terms 5x + 20 = 21x subtract 5x from both sides 20 = 16x divide both sides by 16 1.25 = x

5(1.25) + 26 - 6 = 9(1.25) + 12(1.25) 6.25 + 26 - 6 = 11.25 + 15 26.25 = 26.25

A motorcycle shop maintains an inventory of six times as many new bikes as used bikes. Currently there are 132 new bikes. Use n to represent the unknown number of used bikes.

6n = 132

6x - 4 (6 - x) = 14 6x - 24 + 4x = 14 combine like terms 10x - 24 = 14 add 24 to both sides 10x = 38 divide both sides by 10 x = 3.8

6x - 4 (6 - x) = 14 22.8 - 4 (6 - 3.8) = 14 22.8 - 24 + 15.2 = 14 14 = 14

Which is the SECOND step in solving 1/2x + 6 = -4 ? a. Add 4 to each side of the equation b. Multiply each side of the equation by 2 c. Divide each side of the equation by -4 d. Subtract -6 from each side of the equation

Multiply each side of the equation by 2

The quotient of twelve and a number is one-ninth.

12 / n = 1/9

x/4 = 11 multiply both sides by 4 x = 44

44/4 = 11 11 = 11

Is 4 a solution of 6(x - 9) = -30 ?

Yes 6(4) + 6(-9) = -30 24 - 54 = -30 -30 = -30

Two more than two times a number is five.

2 + 2n = 5

2x + 11 = 1 subtract 11 from both sides 2x = -10 divide both sides by 2 x = -5

2(-5) + 11 = 1 -10 + 11 = 1 1 = 1

2x + 5x = 28 combine like terms 7x = 28 divide both sides by 7 x = 4

2(4) + 5(4) = 28 8 + 20 = 28 28 = 28

2x + 18 = 3 + 5x - 6 combine like terms 2x + 18 = -3 + 5x add 3 to both sides 2x + 21 = 5x subtract 2x from both sides 21 = 3x divide both sides by 3 7 = x

2(7) + 18 = 3 + 5(7) - 6 14 + 18 = 3 + 35 - 6 32 = 32

3 (-x - 7) = -2 (2x + 5) 3(-1x) + 3(-7) = 2(2x) + 2(5) -3x - 21 = -4x - 10 add 21 to both sides -3x = -4x + 11 add 4x to both sides 1x = 11 x = 11

3 (-11 - 7) = -2 ((2)(11) + 5) -33 - 21 = -2 (22 + 5) -33 - 21 = -44 - 10 -54 = -54

3.5 times a number is seventeen.

3.5n = 17

4 (x - 7) + 12 = 2 (x - 6) (4)(1x) + (4)(-7) + 12 = (2)(1x) + (2)(-6) 4x - 28 + 12 = 2x -12 combine like terms 4x - 16 = 2x - 12 add 16 to both sides 4x = 2x + 4 subtract 2x from both sides 2x = 4 divide both sides by 2 x = 2

4 (2 - 7) + 12 = 2 (2 - 6) 8 - 28 + 12 = 4 - 12 -20 -+12 = 4 - 12 -8 = -8

4 = -7 + 8x add 7 to both sides 11 = 8x divide both sides by 8 (11/8) or 1.375 = x

4 = -7 + 8(1.375) 4 = -7 + 11 4 = 4

4x - 9 = 2x + 19 add 9 to both sides 4x = 2x + 28 subtract 2x from both sides 2x = 28 divide both sides by 2 x = 14

4(14) - 9 = 2(14) + 19 56 - 9 = 28 + 19 47 = 47

4x + 15 = 5x + 13 subtract 15 from both sides 4x = 5x - 2 subtract 5x from both sides -1x = -2 divide both sides by -1 x = 2

4(2) + 15 = 5(2) + 13 8 + 15 = 10 + 13 23 = 23

Erica and Steven played a video game. Erica scored 8 less than 4 times Steven's score. Erica's score was 1,000 points. Let x = the number of points Steven scored. How many points did Steven score? a. 4x + 8 = 1,000 b. 1,000 - 4x = 8 c. 8 - 4x = 1,000 d. 4x - 8 = 1,000

4x - 8 = 1,000

5 (2x - 1) + 2 = -2 (5)(2x) + (5)(-1) + 2 = -2 10x - 5 + 2 = -2 combine like terms 10x - 3 = -2 add 3 to both sides 10x = 1 divide both sides by 10 x = 1/10

5 ((2)(1/10) - 1) + 2 = -2 5 (1/5 - 1) + 2 = -2 1 - 5 + 2 = -2 -2 = -2

The sum of five and a number is thirty-nine.

5 + n = 39

5x - 6 - 3x = 3x - 5 combine like terms 2x - 6 = 3x - 5 subtract 5 from both sides 2x - 1 = 3x subtract 2x from both sides -1 = x

5(-1) - 6 - 3(-1) = 3(-1) - 5 -5 - 6 + 3 = -3 - 5 -8 = -8

Is 4 a solution of x - 8 = 2 ?

No 4 - 8 = -4 -4 ≠ 2

-1/2x + 10 = 16 subtract 10 from both sides -1/2x = 6 multiply both sides by -2/1 x = -12

-1/2(-12) + 10 = 16 6 + 10 = 16 16 = 16

-12x - 4x = -3 combine like terms -16x = -3 divide both sides by -16 x = 0.1875

-12(0.1875) - 4(0.1875) = -3 -2.25 - 0.75 = -3 -3 = -3

In the list of numbers, find the one that is a solution of the given equation. 0, -5, -10 x + 5 = 0

-5

-5x + 4 = -4 (2x + 5) -5x + 4 = -8x - 20 add 8x to both sides 3x + 4 = -20 subtract 4 from both sides 3x = -24 divide both sides by 3 x = -8

-5x + 4 = -4 (2x + 5) (-5)(-8) + 4 = -4 ((2)(-8) + 5) 40 + 4 = -4 (-16 + 5) 44 = (-4)(-16) + (-4)(5) 44 = 64 - 20 44 = 44

-6x = -32 + 2x subtract 2x from both sides -8x = -32 divide both sides by -8 x = 4

-6(4) = -32 + 2(4) 24 = -32 + 8 24 = 24

-8x + 7 = 31 subtract 7 from both sides -8x = 24 divide both sides by -8 x = -3

-8(-3) + 7 = 31 24 + 7 = 31 31 = 31

-8x = 40 divide both sides by -8 x = -5

-8(-5) = 40 40 = 40

12 = 5x - 8 add 8 to both sides 20 = 5x divide both sides by 5 4 = x

12 = 5(4) - 8 12 = 20 - 8 12 = 12

56 - 9x = 11x add 9x to both sides 56 = 20x divide both sides by 20 2.8 = x

56 - 9(2.8) = 11(2.8) 56 - 25.2 = 30.8 30.8 = 30.8

What is the CORRECT application of the Distributive Property for 5 (x - 3) - 2 (x + 1) = 15? a. 5x - 3 - 2x + 1 = 15 b. 5x - 15 - 2x - 2 = 15 c. 5x - 15x - 2x - 2 = 15 d. 5x + 15 - 2x + 2 = 15

5x - 15 - 2x - 2 = 15

6 (x - 5) + 4 = 4 (x - 4) 6x -30 + 4 = 4x - 16 combine like terms 6x - 26 = 4x - 16 subtract 4x from both sides 2x - 26 = +16 add 26 to both sides 2x - 10 divide both sides by 2 x = 5

6 (x - 5) + 4 = 4 (x - 4) 30 - 30 + 4 = 20 -16 4 = 4

9x = 8 + 7x subtract 7x from both sides 2x = 8 divide both sides by 2 x = 4

9(4) = 8 + 7(4) 36 = 8 + 14 36 = 36

9x = 6x + 15 subtract 6x from both sides 3x = 15 divide both sides by 3 x = 5

9(5) = 6(5) + 15 45 = 30 + 15 45 = 45

Is -3 a solution for the equation 7x - 2 = 5 ? 7(-3) - 2 = 5 -21 - 2 = -23

No

Is 8 a solution of 2x + 9(x - 4) = 52 ?

Yes 2(8) + 9(8) + 9(-4) = 52 16 + 72 - 36 = 52 88 - 36 = 52 52 = 52

Solution

a solution of an equation is the number(s) that, when substituted for the variable(s), makes the equation true.

Equation

is a mathematical statement that two expressions are equal. All equations contain an equal sign ( = )

The sum of a number and twenty is negative eleven.

n + 20 = -11

A number decreased by six is seventeen.

n - 6 = 17

Five equals the quotient of a number and seven.

n / 7 = 5