Two-step equations involving fractions worksheet with eight problems to solve.

Problem: Solve each two-step equation involving fractions.

We will solve each equation step by step. Let's go through them one by one.

---

#### 1) \( 5a + \frac{2}{3} = -\frac{1}{6} \)

Step 1: Isolate the term with the variable \( a \). Subtract \( \frac{2}{3} \) from both sides.
\[
5a + \frac{2}{3} - \frac{2}{3} = -\frac{1}{6} - \frac{2}{3}
\]
\[
5a = -\frac{1}{6} - \frac{2}{3}
\]

Step 2: Find a common denominator to subtract the fractions on the right-hand side. The common denominator of 6 and 3 is 6.
\[
-\frac{1}{6} - \frac{2}{3} = -\frac{1}{6} - \frac{4}{6} = -\frac{5}{6}
\]
So,
\[
5a = -\frac{5}{6}
\]

Step 3: Solve for \( a \) by dividing both sides by 5.
\[
a = \frac{-\frac{5}{6}}{5} = -\frac{5}{6} \cdot \frac{1}{5} = -\frac{5}{30} = -\frac{1}{6}
\]

Answer:
\[
a = -\frac{1}{6}
\]

---

#### 2) \( \frac{4}{7}(c - \frac{6}{7}) = 8 \)

Step 1: Distribute \( \frac{4}{7} \) across the terms inside the parentheses.
\[
\frac{4}{7} \cdot c - \frac{4}{7} \cdot \frac{6}{7} = 8
\]
\[
\frac{4}{7}c - \frac{24}{49} = 8
\]

Step 2: Isolate the term with the variable \( c \). Add \( \frac{24}{49} \) to both sides.
\[
\frac{4}{7}c - \frac{24}{49} + \frac{24}{49} = 8 + \frac{24}{49}
\]
\[
\frac{4}{7}c = 8 + \frac{24}{49}
\]

Step 3: Convert 8 to a fraction with a denominator of 49.
\[
8 = \frac{392}{49}
\]
So,
\[
\frac{4}{7}c = \frac{392}{49} + \frac{24}{49} = \frac{416}{49}
\]

Step 4: Solve for \( c \) by multiplying both sides by the reciprocal of \( \frac{4}{7} \), which is \( \frac{7}{4} \).
\[
c = \frac{416}{49} \cdot \frac{7}{4} = \frac{416 \cdot 7}{49 \cdot 4} = \frac{2912}{196} = \frac{1456}{98} = \frac{728}{49} = 14.857 \approx 14\frac{6}{7}
\]

Answer:
\[
c = 14\frac{6}{7}
\]

---

#### 3) \( \frac{r - 1}{2} = 0 \)

Step 1: Eliminate the denominator by multiplying both sides by 2.
\[
2 \cdot \frac{r - 1}{2} = 2 \cdot 0
\]
\[
r - 1 = 0
\]

Step 2: Solve for \( r \) by adding 1 to both sides.
\[
r - 1 + 1 = 0 + 1
\]
\[
r = 1
\]

Answer:
\[
r = 1
\]

---

#### 4) \( -10 - m = -\frac{5}{9} \)

Step 1: Isolate the term with the variable \( m \). Add 10 to both sides.
\[
-10 - m + 10 = -\frac{5}{9} + 10
\]
\[
-m = -\frac{5}{9} + 10
\]

Step 2: Convert 10 to a fraction with a denominator of 9.
\[
10 = \frac{90}{9}
\]
So,
\[
-m = -\frac{5}{9} + \frac{90}{9} = \frac{85}{9}
\]

Step 3: Solve for \( m \) by multiplying both sides by -1.
\[
m = -\frac{85}{9}
\]

Answer:
\[
m = -\frac{85}{9}
\]

---

#### 5) \( \frac{6}{5} = \frac{3}{5}u + \frac{6}{5} \)

Step 1: Isolate the term with the variable \( u \). Subtract \( \frac{6}{5} \) from both sides.
\[
\frac{6}{5} - \frac{6}{5} = \frac{3}{5}u + \frac{6}{5} - \frac{6}{5}
\]
\[
0 = \frac{3}{5}u
\]

Step 2: Solve for \( u \). Since \( \frac{3}{5}u = 0 \), it follows that:
\[
u = 0
\]

Answer:
\[
u = 0
\]

---

#### 6) \( \frac{h + 4}{3} = \frac{2}{9} \)

Step 1: Eliminate the denominator by multiplying both sides by 3.
\[
3 \cdot \frac{h + 4}{3} = 3 \cdot \frac{2}{9}
\]
\[
h + 4 = \frac{6}{9} = \frac{2}{3}
\]

Step 2: Isolate \( h \) by subtracting 4 from both sides.
\[
h + 4 - 4 = \frac{2}{3} - 4
\]
\[
h = \frac{2}{3} - 4
\]

Step 3: Convert 4 to a fraction with a denominator of 3.
\[
4 = \frac{12}{3}
\]
So,
\[
h = \frac{2}{3} - \frac{12}{3} = -\frac{10}{3}
\]

Answer:
\[
h = -\frac{10}{3}
\]

---

#### 7) \( \frac{1}{6}k - 1 = 5 \)

Step 1: Isolate the term with the variable \( k \). Add 1 to both sides.
\[
\frac{1}{6}k - 1 + 1 = 5 + 1
\]
\[
\frac{1}{6}k = 6
\]

Step 2: Solve for \( k \) by multiplying both sides by 6.
\[
k = 6 \cdot 6 = 36
\]

Answer:
\[
k = 36
\]

---

#### 8) \( -\frac{1}{8} = -\frac{7}{4} + \frac{3}{2}z \)

Step 1: Isolate the term with the variable \( z \). Add \( \frac{7}{4} \) to both sides.
\[
-\frac{1}{8} + \frac{7}{4} = -\frac{7}{4} + \frac{7}{4} + \frac{3}{2}z
\]
\[
-\frac{1}{8} + \frac{7}{4} = \frac{3}{2}z
\]

Step 2: Find a common denominator to add the fractions on the left-hand side. The common denominator of 8 and 4 is 8.
\[
-\frac{1}{8} + \frac{7}{4} = -\frac{1}{8} + \frac{14}{8} = \frac{13}{8}
\]
So,
\[
\frac{13}{8} = \frac{3}{2}z
\]

Step 3: Solve for \( z \) by multiplying both sides by the reciprocal of \( \frac{3}{2} \), which is \( \frac{2}{3} \).
\[
z = \frac{13}{8} \cdot \frac{2}{3} = \frac{13 \cdot 2}{8 \cdot 3} = \frac{26}{24} = \frac{13}{12}
\]

Answer:
\[
z = \frac{13}{12}
\]

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
1) & a = -\frac{1}{6} \\
2) & c = 14\frac{6}{7} \\
3) & r = 1 \\
4) & m = -\frac{85}{9} \\
5) & u = 0 \\
6) & h = -\frac{10}{3} \\
7) & k = 36 \\
8) & z = \frac{13}{12}
\end{array}
}
\]