Solve each equation and show all your work. 6x - 5 = 2x + 7

3z - 2 = 5z + 19

5x - 15 + 2x = 4x + 12

35x + 2 = 7

x = 3

z = (-21)/2

x = 9

x = 1/7

Step-by-step explanation:

Solve for x:

6 x - 5 = 2 x + 7

Subtract 2 x from both sides:

(6 x - 2 x) - 5 = (2 x - 2 x) + 7

6 x - 2 x = 4 x:

4 x - 5 = (2 x - 2 x) + 7

2 x - 2 x = 0:

4 x - 5 = 7

Add 5 to both sides:

4 x + (5 - 5) = 5 + 7

5 - 5 = 0:

4 x = 7 + 5

7 + 5 = 12:

4 x = 12

Divide both sides of 4 x = 12 by 4:

(4 x)/4 = 12/4

4/4 = 1:

x = 12/4

The gcd of 12 and 4 is 4, so 12/4 = (4×3)/(4×1) = 4/4×3 = 3:

x = 3

Solve for z:

3 z - 2 = 5 z + 19

Subtract 5 z from both sides:

(3 z - 5 z) - 2 = (5 z - 5 z) + 19

3 z - 5 z = -2 z:

-2 z - 2 = (5 z - 5 z) + 19

5 z - 5 z = 0:

-2 z - 2 = 19

Add 2 to both sides:

(2 - 2) - 2 z = 2 + 19

2 - 2 = 0:

-2 z = 19 + 2

19 + 2 = 21:

-2 z = 21

Divide both sides of -2 z = 21 by -2:

(-2 z)/(-2) = 21/(-2)

(-2)/(-2) = 1:

z = 21/(-2)

Multiply numerator and denominator of 21/(-2) by -1:

z = (-21)/2

Solve for x:

5 x + 2 x - 15 = 4 x + 12

Grouping like terms, 5 x + 2 x - 15 = (5 x + 2 x) - 15:

(5 x + 2 x) - 15 = 4 x + 12

5 x + 2 x = 7 x:

7 x - 15 = 4 x + 12

Subtract 4 x from both sides:

(7 x - 4 x) - 15 = (4 x - 4 x) + 12

7 x - 4 x = 3 x:

3 x - 15 = (4 x - 4 x) + 12

4 x - 4 x = 0:

3 x - 15 = 12

Add 15 to both sides:

3 x + (15 - 15) = 15 + 12

15 - 15 = 0:

3 x = 12 + 15

12 + 15 = 27:

3 x = 27

Divide both sides of 3 x = 27 by 3:

(3 x)/3 = 27/3

3/3 = 1:

x = 27/3

The gcd of 27 and 3 is 3, so 27/3 = (3×9)/(3×1) = 3/3×9 = 9:

x = 9

Solve for x:

35 x + 2 = 7

Subtract 2 from both sides:

35 x + (2 - 2) = 7 - 2

2 - 2 = 0:

35 x = 7 - 2

7 - 2 = 5:

35 x = 5

Divide both sides of 35 x = 5 by 35:

(35 x)/35 = 5/35

35/35 = 1:

x = 5/35

The gcd of 5 and 35 is 5, so 5/35 = (5×1)/(5×7) = 5/5×1/7 = 1/7:

x = 1/7

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step-by-step explanation: