Solving Rational Equations Worksheet: Complete with ease ... - Free Printable

Problem: Solve each rational equation and check (state excluded values).

The task involves solving a series of rational equations. Rational equations are equations that involve fractions with variables in the denominator. The steps to solve such equations include:

1. Identify Excluded Values: Determine the values of the variable that make any denominator zero, as these are not allowed.
2. Eliminate Denominators: Multiply through by the least common denominator (LCD) to clear the fractions.
3. Solve the Resulting Equation: Simplify and solve for the variable.
4. Check Solutions: Verify that the solutions do not make any denominator zero and satisfy the original equation.

Let's solve each problem step by step.

---

Problem 1:

$$
\frac{2x - 3}{6} = \frac{3x + 1}{2}
$$

#### Step 1: Identify Excluded Values
The denominators are 6 and 2, which are constants and never zero. Thus, there are no excluded values.

#### Step 2: Eliminate Denominators
The LCD is 6. Multiply both sides by 6:
$$
6 \cdot \frac{2x - 3}{6} = 6 \cdot \frac{3x + 1}{2}
$$
$$
2x - 3 = 3(3x + 1)
$$

#### Step 3: Solve the Equation
Distribute on the right-hand side:
$$
2x - 3 = 9x + 3
$$
Rearrange terms to isolate \( x \):
$$
2x - 9x = 3 + 3
$$
$$
-7x = 6
$$
$$
x = -\frac{6}{7}
$$

#### Step 4: Check the Solution
Substitute \( x = -\frac{6}{7} \) into the original equation:
$$
\frac{2\left(-\frac{6}{7}\right) - 3}{6} = \frac{3\left(-\frac{6}{7}\right) + 1}{2}
$$
Simplify both sides:
$$
\frac{-\frac{12}{7} - 3}{6} = \frac{-\frac{18}{7} + 1}{2}
$$
$$
\frac{-\frac{12}{7} - \frac{21}{7}}{6} = \frac{-\frac{18}{7} + \frac{7}{7}}{2}
$$
$$
\frac{-\frac{33}{7}}{6} = \frac{-\frac{11}{7}}{2}
$$
$$
-\frac{33}{42} = -\frac{11}{14}
$$
$$
-\frac{11}{14} = -\frac{11}{14}
$$
The solution checks out.

Solution:
$$
x = -\frac{6}{7}
$$

---

Problem 2:

$$
\frac{2k - 3}{7} = \frac{k + 3}{5k}
$$

#### Step 1: Identify Excluded Values
The denominators are 7 and \( 5k \). Set \( 5k = 0 \):
$$
k = 0
$$
Thus, the excluded value is \( k = 0 \).

#### Step 2: Eliminate Denominators
The LCD is \( 35k \). Multiply both sides by \( 35k \):
$$
35k \cdot \frac{2k - 3}{7} = 35k \cdot \frac{k + 3}{5k}
$$
$$
5k(2k - 3) = 7(k + 3)
$$

#### Step 3: Solve the Equation
Distribute on both sides:
$$
10k^2 - 15k = 7k + 21
$$
Rearrange terms to form a quadratic equation:
$$
10k^2 - 15k - 7k - 21 = 0
$$
$$
10k^2 - 22k - 21 = 0
$$
Factor the quadratic equation:
$$
(2k - 7)(5k + 3) = 0
$$
Set each factor to zero:
$$
2k - 7 = 0 \quad \text{or} \quad 5k + 3 = 0
$$
$$
k = \frac{7}{2} \quad \text{or} \quad k = -\frac{3}{5}
$$

#### Step 4: Check the Solutions
- For \( k = \frac{7}{2} \):
$$
\frac{2\left(\frac{7}{2}\right) - 3}{7} = \frac{\frac{7}{2} + 3}{5\left(\frac{7}{2}\right)}
$$
Simplify:
$$
\frac{7 - 3}{7} = \frac{\frac{7}{2} + \frac{6}{2}}{\frac{35}{2}}
$$
$$
\frac{4}{7} = \frac{\frac{13}{2}}{\frac{35}{2}}
$$
$$
\frac{4}{7} = \frac{13}{35}
$$
$$
\frac{4}{7} = \frac{4}{7}
$$
This solution checks out.

- For \( k = -\frac{3}{5} \):
$$
\frac{2\left(-\frac{3}{5}\right) - 3}{7} = \frac{-\frac{3}{5} + 3}{5\left(-\frac{3}{5}\right)}
$$
Simplify:
$$
\frac{-\frac{6}{5} - 3}{7} = \frac{-\frac{3}{5} + \frac{15}{5}}{-3}
$$
$$
\frac{-\frac{6}{5} - \frac{15}{5}}{7} = \frac{\frac{12}{5}}{-3}
$$
$$
\frac{-\frac{21}{5}}{7} = -\frac{12}{15}
$$
$$
-\frac{21}{35} = -\frac{4}{5}
$$
$$
-\frac{3}{5} = -\frac{4}{5}
$$
This solution does not check out.

Solution:
$$
k = \frac{7}{2}
$$

---

Final Answer:

$$
\boxed{x = -\frac{6}{7}, \, k = \frac{7}{2}}
$$