Let's solve each of these equations step by step using
cross-multiplication, which is a method used to solve proportions (equations with fractions). The rule is:
> If $\frac{a}{b} = \frac{c}{d}$, then $a \cdot d = b \cdot c$.
We'll go through each problem one at a time.
---
1. $\frac{x}{7} = \frac{3}{2}$
Cross-multiply:
$$
x \cdot 2 = 7 \cdot 3
\Rightarrow 2x = 21
$$
Solve for $x$:
$$
x = \frac{21}{2} = 10.5
$$
✔ Answer: $x = \frac{21}{2}$ or $10.5$
---
2. $\frac{2x}{7} = \frac{3}{2}$
Cross-multiply:
$$
(2x) \cdot 2 = 7 \cdot 3
\Rightarrow 4x = 21
$$
Solve for $x$:
$$
x = \frac{21}{4} = 5.25
$$
✔ Answer: $x = \frac{21}{4}$
---
3. $\frac{2x}{7} = \frac{3}{14}$
Cross-multiply:
$$
(2x) \cdot 14 = 7 \cdot 3
\Rightarrow 28x = 21
$$
Solve for $x$:
$$
x = \frac{21}{28} = \frac{3}{4}
$$
✔ Answer: $x = \frac{3}{4}$
---
4. $\frac{2x}{14} = \frac{3}{14}$
Note: Both sides have the same denominator, so we can simplify or cross-multiply.
Cross-multiply:
$$
(2x) \cdot 14 = 14 \cdot 3
\Rightarrow 28x = 42
$$
Solve for $x$:
$$
x = \frac{42}{28} = \frac{3}{2}
$$
✔ Answer: $x = \frac{3}{2}$
Alternatively, notice that both sides have denominator 14, so:
$$
\frac{2x}{14} = \frac{3}{14} \Rightarrow 2x = 3 \Rightarrow x = \frac{3}{2}
$$
Same result.
---
7. $\frac{14}{2x+1} = \frac{7}{3}$
Cross-multiply:
$$
14 \cdot 3 = 7 \cdot (2x + 1)
\Rightarrow 42 = 14x + 7
$$
Solve for $x$:
$$
42 - 7 = 14x \Rightarrow 35 = 14x
\Rightarrow x = \frac{35}{14} = \frac{5}{2}
$$
✔ Answer: $x = \frac{5}{2}$
---
8. $\frac{14}{2x-1} = \frac{7}{3}$
Cross-multiply:
$$
14 \cdot 3 = 7 \cdot (2x - 1)
\Rightarrow 42 = 14x - 7
$$
Solve for $x$:
$$
42 + 7 = 14x \Rightarrow 49 = 14x
\Rightarrow x = \frac{49}{14} = \frac{7}{2}
$$
✔ Answer: $x = \frac{7}{2}$
---
9. $\frac{7}{2x-1} = \frac{7}{3}$
Cross-multiply:
$$
7 \cdot 3 = 7 \cdot (2x - 1)
\Rightarrow 21 = 14x - 7
$$
Solve for $x$:
$$
21 + 7 = 14x \Rightarrow 28 = 14x
\Rightarrow x = \frac{28}{14} = 2
$$
✔ Answer: $x = 2$
---
10. $\frac{2x - 1}{7} = \frac{7}{3}$
Cross-multiply:
$$
(2x - 1) \cdot 3 = 7 \cdot 7
\Rightarrow 6x - 3 = 49
$$
Solve for $x$:
$$
6x = 49 + 3 = 52
\Rightarrow x = \frac{52}{6} = \frac{26}{3}
$$
✔ Answer: $x = \frac{26}{3}$
---
✔ Final Answers Summary:
| Problem | Answer |
|--------|--------|
| 1. | $x = \frac{21}{2}$ |
| 2. | $x = \frac{21}{4}$ |
| 3. | $x = \frac{3}{4}$ |
| 4. | $x = \frac{3}{2}$ |
| 7. | $x = \frac{5}{2}$ |
| 8. | $x = \frac{7}{2}$ |
| 9. | $x = 2$ |
| 10. | $x = \frac{26}{3}$ |
Let me know if you'd like these checked graphically or explained further!
Parent Tip: Review the logic above to help your child master the concept of solving rational equations by cross multiplying worksheet.
