Solving Equations with fractions. - Free Printable

Let's solve each equation step-by-step. We'll go through Q1 and Q2 systematically, using algebraic methods to solve equations involving fractions.

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Q1: Solve the following equations

#### [a] $\frac{2x + 3}{x - 1} = 7$

Step 1: Multiply both sides by $x - 1$ (assuming $x \ne 1$):

$$
2x + 3 = 7(x - 1)
$$

Step 2: Expand the right side:

$$
2x + 3 = 7x - 7
$$

Step 3: Bring all terms to one side:

$$
3 + 7 = 7x - 2x \Rightarrow 10 = 5x
$$

Step 4: Solve for $x$:

$$
x = \frac{10}{5} = 2
$$

✔ Check: $x = 2$, denominator $x - 1 = 1 \ne 0$, so valid.

> Answer: $x = 2$

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#### [b] $\frac{4x - 2}{x + 5} = 2$

Step 1: Multiply both sides by $x + 5$ (assume $x \ne -5$):

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

Step 2: Expand:

$$
4x - 2 = 2x + 10
$$

Step 3: Solve:

$$
4x - 2x = 10 + 2 \Rightarrow 2x = 12 \Rightarrow x = 6
$$

✔ Check: $x = 6$, denominator $x + 5 = 11 \ne 0$, valid.

> Answer: $x = 6$

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#### [c] $\frac{5x - 3}{x - 2} = 4$

Step 1: Multiply both sides by $x - 2$ ($x \ne 2$):

$$
5x - 3 = 4(x - 2)
$$

Step 2: Expand:

$$
5x - 3 = 4x - 8
$$

Step 3: Solve:

$$
5x - 4x = -8 + 3 \Rightarrow x = -5
$$

✔ Check: $x = -5$, denominator $x - 2 = -7 \ne 0$, valid.

> Answer: $x = -5$

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#### [d] $\frac{3x - 3}{x + 3} = 6$

Step 1: Multiply both sides by $x + 3$ ($x \ne -3$):

$$
3x - 3 = 6(x + 3)
$$

Step 2: Expand:

$$
3x - 3 = 6x + 18
$$

Step 3: Solve:

$$
-3 - 18 = 6x - 3x \Rightarrow -21 = 3x \Rightarrow x = -7
$$

✔ Check: $x = -7$, denominator $x + 3 = -4 \ne 0$, valid.

> Answer: $x = -7$

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#### [e] $\frac{5x + 2}{x - 10} = 1$

Step 1: Multiply both sides by $x - 10$ ($x \ne 10$):

$$
5x + 2 = 1(x - 10)
$$

Step 2: Expand:

$$
5x + 2 = x - 10
$$

Step 3: Solve:

$$
5x - x = -10 - 2 \Rightarrow 4x = -12 \Rightarrow x = -3
$$

✔ Check: $x = -3$, denominator $x - 10 = -13 \ne 0$, valid.

> Answer: $x = -3$

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Q2: Solve the following equations

These are equations with two fractions equal to each other. We use cross-multiplication.

#### [a] $\frac{3}{x + 2} = \frac{1}{x + 4}$

Step 1: Cross-multiply:

$$
3(x + 4) = 1(x + 2)
$$

Step 2: Expand:

$$
3x + 12 = x + 2
$$

Step 3: Solve:

$$
3x - x = 2 - 12 \Rightarrow 2x = -10 \Rightarrow x = -5
$$

✔ Check: $x = -5$, denominators: $x+2 = -3$, $x+4 = -1$, both ≠ 0 → valid.

> Answer: $x = -5$

---

#### [b] $\frac{4}{x - 2} = \frac{2}{x - 4}$

Step 1: Cross-multiply:

$$
4(x - 4) = 2(x - 2)
$$

Step 2: Expand:

$$
4x - 16 = 2x - 4
$$

Step 3: Solve:

$$
4x - 2x = -4 + 16 \Rightarrow 2x = 12 \Rightarrow x = 6
$$

✔ Check: $x = 6$, denominators: $x - 2 = 4$, $x - 4 = 2$, both ≠ 0 → valid.

> Answer: $x = 6$

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#### [c] $\frac{1}{x - 3} = \frac{2}{x + 6}$

Step 1: Cross-multiply:

$$
1(x + 6) = 2(x - 3)
$$

Step 2: Expand:

$$
x + 6 = 2x - 6
$$

Step 3: Solve:

$$
6 + 6 = 2x - x \Rightarrow 12 = x
$$

✔ Check: $x = 12$, denominators: $x - 3 = 9$, $x + 6 = 18$, both ≠ 0 → valid.

> Answer: $x = 12$

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#### [d] $\frac{2}{x + 1} = \frac{3}{x - 1}$

Step 1: Cross-multiply:

$$
2(x - 1) = 3(x + 1)
$$

Step 2: Expand:

$$
2x - 2 = 3x + 3
$$

Step 3: Solve:

$$
-2 - 3 = 3x - 2x \Rightarrow -5 = x
$$

✔ Check: $x = -5$, denominators: $x + 1 = -4$, $x - 1 = -6$, both ≠ 0 → valid.

> Answer: $x = -5$

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#### [e] $\frac{5}{x + 1} = \frac{4}{x - 4}$

Step 1: Cross-multiply:

$$
5(x - 4) = 4(x + 1)
$$

Step 2: Expand:

$$
5x - 20 = 4x + 4
$$

Step 3: Solve:

$$
5x - 4x = 4 + 20 \Rightarrow x = 24
$$

✔ Check: $x = 24$, denominators: $x + 1 = 25$, $x - 4 = 20$, both ≠ 0 → valid.

> Answer: $x = 24$

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✔ Final Answers Summary

#### Q1:
- [a] $x = 2$
- [b] $x = 6$
- [c] $x = -5$
- [d] $x = -7$
- [e] $x = -3$

#### Q2:
- [a] $x = -5$
- [b] $x = 6$
- [c] $x = 12$
- [d] $x = -5$
- [e] $x = 24$

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🔍 Key Tips:

- Always check for restrictions (values that make denominator zero).
- Use cross-multiplication when two fractions are equal.
- After solving, always verify your answer in the original equation.

Let me know if you'd like a visual explanation or graph! 😊