Question

3.08: variables on both sides of the equation
use notebook paper. you must write the original equation on your paper.
solve each equation.
work must be shown going down under the original equation. each step is written under the previous line of work. each line may contain only one equal sign.
you must show inverse operations on both sides of the equation.
all mixed numbers are to be converted to improper fraction. all fractions must be simplified, including signs, and left improper when applicable. decimals may only be used when the original equation contains a decimal.

1. 10 + 8c + 15 = -2c - 40
2. 15y + 14 = 2(5y + 6)
3. \\(\frac{1}{2}(6x - 4) = 4x - 9\\)
4. 5(5a + 3) = 14a - 29
5. 8(k + 6) = -3(k - 33)
6. 4 - 6x = 60 - 2x
7. 17 - 5a + 9a = -12 + 6a -1
8. -3(3m - 10) - 7 = -5(m + 1) + 3m
9. 6x + 34 = 3 - (2x + 17)
10. -2(6y - 1) = -\\(\frac{5}{3}\\)(3y + 15) + 6

11/12:
the polygons below have the same perimeter. find the value of x and then find of the polygons.

Explanation:

Let's solve equation 1: \(10 + 8c + 15 = -2c - 40\)

Step1: Combine like terms on left

\(10 + 15 + 8c = -2c - 40\)
\(25 + 8c = -2c - 40\)

Step2: Add \(2c\) to both sides

\(25 + 8c + 2c = -2c + 2c - 40\)
\(25 + 10c = -40\)

Step3: Subtract 25 from both sides

\(25 - 25 + 10c = -40 - 25\)
\(10c = -65\)

Step4: Divide by 10

\(\frac{10c}{10} = \frac{-65}{10}\)
\(c = -\frac{13}{2}\) (or \(c = -6.5\))

Now equation 2: \(15y + 14 = 2(5y + 6)\)

Step1: Distribute right side

\(15y + 14 = 10y + 12\)

Step2: Subtract \(10y\) from both sides

\(15y - 10y + 14 = 10y - 10y + 12\)
\(5y + 14 = 12\)

Step3: Subtract 14 from both sides

\(5y + 14 - 14 = 12 - 14\)
\(5y = -2\)

Step4: Divide by 5

\(\frac{5y}{5} = \frac{-2}{5}\)
\(y = -\frac{2}{5}\)

Equation 3: \(\frac{1}{2}(6x - 4) = 4x - 9\)

Step1: Distribute left side

\(3x - 2 = 4x - 9\)

Step2: Subtract \(3x\) from both sides

\(3x - 3x - 2 = 4x - 3x - 9\)
\(-2 = x - 9\)

Step3: Add 9 to both sides

\(-2 + 9 = x - 9 + 9\)
\(7 = x\) (or \(x = 7\))

Equation 4: \(5(5a + 3) = 14a - 29\)

Step1: Distribute left side

\(25a + 15 = 14a - 29\)

Step2: Subtract \(14a\) from both sides

\(25a - 14a + 15 = 14a - 14a - 29\)
\(11a + 15 = -29\)

Step3: Subtract 15 from both sides

\(11a + 15 - 15 = -29 - 15\)
\(11a = -44\)

Step4: Divide by 11

\(\frac{11a}{11} = \frac{-44}{11}\)
\(a = -4\)

Equation 5: \(8(k + 6) = -3(k - 33)\)

Step1: Distribute both sides

\(8k + 48 = -3k + 99\)

Step2: Add \(3k\) to both sides

\(8k + 3k + 48 = -3k + 3k + 99\)
\(11k + 48 = 99\)

Step3: Subtract 48 from both sides

\(11k + 48 - 48 = 99 - 48\)
\(11k = 51\)

Step4: Divide by 11

\(\frac{11k}{11} = \frac{51}{11}\)
\(k = \frac{51}{11}\) (or \(k \approx 4.636\))

Equation 6: \(4 - 6x = 60 - 2x\)

Step1: Add \(6x\) to both sides

\(4 - 6x + 6x = 60 - 2x + 6x\)
\(4 = 60 + 4x\)

Step2: Subtract 60 from both sides

\(4 - 60 = 60 - 60 + 4x\)
\(-56 = 4x\)

Step3: Divide by 4

\(\frac{-56}{4} = \frac{4x}{4}\)
\(-14 = x\) (or \(x = -14\))

Equation 7: \(17 - 5a + 9a = -12 + 6a - 1\)

Step1: Combine like terms

Left: \(17 + 4a\), Right: \(-13 + 6a\)
\(17 + 4a = -13 + 6a\)

Step2: Subtract \(4a\) from both sides

\(17 + 4a - 4a = -13 + 6a - 4a\)
\(17 = -13 + 2a\)

Step3: Add 13 to both sides

\(17 + 13 = -13 + 13 + 2a\)
\(30 = 2a\)

Step4: Divide by 2

\(\frac{30}{2} = \frac{2a}{2}\)
\(15 = a\) (or \(a = 15\))

Equation 8: \(-3(3m - 10) - 7 = -5(m + 1) + 3m\)

Step1: Distribute both sides

Left: \(-9m + 30 - 7 = -9m + 23\)
Right: \(-5m - 5 + 3m = -2m - 5\)
\(-9m + 23 = -2m - 5\)

Step2: Add \(9m\) to both sides

\(-9m + 9m + 23 = -2m + 9m - 5\)
\(23 = 7m - 5\)

Step3: Add 5 to both sides

\(23 + 5 = 7m - 5 + 5\)
\(28 = 7m\)

Step4: Divide by 7

\(\frac{28}{7} = \frac{7m}{7}\)
\(4 = m\) (or \(m = 4\))

Equation 9: \(6x + 34 = 3 - (2x + 17)\)

Step1: Distribute right side (note the negative)

\(6x + 34 = 3 - 2x - 17\)

Step2: Combine like terms on right

\(6x + 34 = -2x - 14\)

Step3: Add \(2x\) to both sides

\(6x + 2x + 34 = -2x + 2x - 14\)
\(8x + 34 = -14\)

Step4: Subtract 34 from both sides

\(8x + 34 - 34 = -14 - 34\)
\(8x = -48\)

Step5: Divide by 8

\(\frac{8x}{8} = \frac{-48}{8}\)
\(x = -6\)

Equation 10: \(-2(6y - 1) = -\frac{5}{3}(3y + 15) + 6\)

Step1: Distribute both sides

Left: \(-12y + 2\)
Right: \(-5y - 25 + 6 = -5y - 19\)
\(-12y + 2 = -5y - 19\)

Step2: Add \(12y\) to both sides

\(-12y + 12y + 2 = -5y + 12y - 19\)
\(2 = 7y - 19\)

Step3: Add 19 to both sides

\(2 + 19 = 7y - 19 + 19\)
\(21 = 7y\)

Step4: Divide by 7

\(\frac{21}{7} = \frac{7y}{7}\)
\(3 = y\) (or \(y = 3\))

For 11/12 (polygons with same perimeter), we need the p…

Answer:

s (for solved equations):

1. \(c = -\frac{13}{2}\)
2. \(y = -\frac{2}{5}\)
3. \(x = 7\)
4. \(a = -4\)
5. \(k = \frac{51}{11}\)
6. \(x = -14\)
7. \(a = 15\)
8. \(m = 4\)
9. \(x = -6\)
10. \(y = 3\)

(For 11/12, provide polygon details to solve for \(x\) and perimeters.)