IW) Solving Linear Equations with Decimals - Interwoven Maths - Free Printable

Let's solve each of these linear equations step by step. I'll go through them one at a time, showing the process and simplifying decimals where necessary.

---

1) $ 0.2x + 5 = 8 $

Subtract 5 from both sides:
$$
0.2x = 3
$$

Divide by 0.2:
$$
x = \frac{3}{0.2} = 15
$$

✔ Answer: $ x = 15 $

---

2) $ 0.2x - 0.5 = 8 $

Add 0.5 to both sides:
$$
0.2x = 8.5
$$

Divide by 0.2:
$$
x = \frac{8.5}{0.2} = 42.5
$$

✔ Answer: $ x = 42.5 $

---

3) $ 0.5x + 0.2 = 8 $

Subtract 0.2:
$$
0.5x = 7.8
$$

Divide by 0.5:
$$
x = \frac{7.8}{0.5} = 15.6
$$

✔ Answer: $ x = 15.6 $

---

4) $ 0.8x - 0.2 = 50 $

Add 0.2:
$$
0.8x = 50.2
$$

Divide by 0.8:
$$
x = \frac{50.2}{0.8} = 62.75
$$

✔ Answer: $ x = 62.75 $

---

5) $ 0.2x + 0.15 = 0.5x $

Subtract $ 0.2x $ from both sides:
$$
0.15 = 0.3x
$$

Divide by 0.3:
$$
x = \frac{0.15}{0.3} = 0.5
$$

✔ Answer: $ x = 0.5 $

---

6) $ 2x + 0.8 = 2 - 0.5x $

Add $ 0.5x $ to both sides:
$$
2.5x + 0.8 = 2
$$

Subtract 0.8:
$$
2.5x = 1.2
$$

Divide by 2.5:
$$
x = \frac{1.2}{2.5} = 0.48
$$

✔ Answer: $ x = 0.48 $

---

7) $ 0.002x + 50 = 80 $

Subtract 50:
$$
0.002x = 30
$$

Divide by 0.002:
$$
x = \frac{30}{0.002} = 15,000
$$

✔ Answer: $ x = 15,000 $

---

8) $ 80x + 0.002 = 0.05 $

Subtract 0.002:
$$
80x = 0.048
$$

Divide by 80:
$$
x = \frac{0.048}{80} = 0.0006
$$

✔ Answer: $ x = 0.0006 $

---

9) $ 0.2x - 0.07 = 0.35 $

Add 0.07:
$$
0.2x = 0.42
$$

Divide by 0.2:
$$
x = \frac{0.42}{0.2} = 2.1
$$

✔ Answer: $ x = 2.1 $

---

10) $ 0.03x + 0.04 = 0.46 $

Subtract 0.04:
$$
0.03x = 0.42
$$

Divide by 0.03:
$$
x = \frac{0.42}{0.03} = 14
$$

✔ Answer: $ x = 14 $

---

11) $ 9.7 - 0.07x = 1.3 $

Subtract 9.7:
$$
-0.07x = 1.3 - 9.7 = -8.4
$$

Divide by -0.07:
$$
x = \frac{-8.4}{-0.07} = 120
$$

✔ Answer: $ x = 120 $

---

12) $ 3.3 + 0.03x = 0.07x + 0.5 $

Subtract $ 0.03x $ and 0.5 from both sides:
$$
3.3 - 0.5 = 0.07x - 0.03x \Rightarrow 2.8 = 0.04x
$$

Divide by 0.04:
$$
x = \frac{2.8}{0.04} = 70
$$

✔ Answer: $ x = 70 $

---

13) $ 0.2(x + 0.8) = 0.19 $

Distribute:
$$
0.2x + 0.16 = 0.19
$$

Subtract 0.16:
$$
0.2x = 0.03
$$

Divide by 0.2:
$$
x = \frac{0.03}{0.2} = 0.15
$$

✔ Answer: $ x = 0.15 $

---

14) $ 0.4(2x - 0.03) = 0.3 $

Distribute:
$$
0.8x - 0.012 = 0.3
$$

Add 0.012:
$$
0.8x = 0.312
$$

Divide by 0.8:
$$
x = \frac{0.312}{0.8} = 0.39
$$

✔ Answer: $ x = 0.39 $

---

15) $ 0.4(0.2x - 0.03) = 0.03 $

Distribute:
$$
0.08x - 0.012 = 0.03
$$

Add 0.012:
$$
0.08x = 0.042
$$

Divide by 0.08:
$$
x = \frac{0.042}{0.08} = 0.525
$$

✔ Answer: $ x = 0.525 $

---

16) $ 0.6(0.7x - 0.81) = 0.9 $

Distribute:
$$
0.42x - 0.486 = 0.9
$$

Add 0.486:
$$
0.42x = 1.386
$$

Divide by 0.42:
$$
x = \frac{1.386}{0.42} = 3.3
$$

✔ Answer: $ x = 3.3 $

---

17) $ \frac{5x - 0.8}{0.2} = 3.5 $

Multiply both sides by 0.2:
$$
5x - 0.8 = 0.7
$$

Add 0.8:
$$
5x = 1.5
$$

Divide by 5:
$$
x = 0.3
$$

✔ Answer: $ x = 0.3 $

---

18) $ \frac{0.9x - 0.04}{0.8} = 0.4 $

Multiply both sides by 0.8:
$$
0.9x - 0.04 = 0.32
$$

Add 0.04:
$$
0.9x = 0.36
$$

Divide by 0.9:
$$
x = \frac{0.36}{0.9} = 0.4
$$

✔ Answer: $ x = 0.4 $

---

19) $ \frac{0.52x - 0.04}{0.8} = 0.4x $

Multiply both sides by 0.8:
$$
0.52x - 0.04 = 0.32x
$$

Subtract $ 0.32x $:
$$
0.2x - 0.04 = 0
$$

Add 0.04:
$$
0.2x = 0.04
$$

Divide by 0.2:
$$
x = \frac{0.04}{0.2} = 0.2
$$

✔ Answer: $ x = 0.2 $

---

20) $ \frac{0.7x - 0.021}{0.3x} = 1.4 $

Multiply both sides by $ 0.3x $:
$$
0.7x - 0.021 = 1.4 \cdot 0.3x = 0.42x
$$

Subtract $ 0.42x $:
$$
0.28x - 0.021 = 0
$$

Add 0.021:
$$
0.28x = 0.021
$$

Divide by 0.28:
$$
x = \frac{0.021}{0.28} = 0.075
$$

✔ Answer: $ x = 0.075 $

---

21) $ \frac{0.13}{0.004x} = 5.2 $

Multiply both sides by $ 0.004x $:
$$
0.13 = 5.2 \cdot 0.004x = 0.0208x
$$

Divide by 0.0208:
$$
x = \frac{0.13}{0.0208} = 6.25
$$

✔ Answer: $ x = 6.25 $

---

22) $ \frac{0.3}{1.4}(0.4x - 0.5) = 4.2 $

First, simplify $ \frac{0.3}{1.4} = \frac{3}{14} \approx 0.2142857 $

But better to keep exact:

$$
\frac{0.3}{1.4}(0.4x - 0.5) = 4.2
$$

Multiply both sides by 1.4:
$$
0.3(0.4x - 0.5) = 4.2 \cdot 1.4 = 5.88
$$

Now distribute:
$$
0.12x - 0.15 = 5.88
$$

Add 0.15:
$$
0.12x = 6.03
$$

Divide by 0.12:
$$
x = \frac{6.03}{0.12} = 50.25
$$

✔ Answer: $ x = 50.25 $

---

23) $ \frac{0.3}{1.4}(19.4x - 0.5) = 4.2x $

Again, $ \frac{0.3}{1.4} = \frac{3}{14} $

So:
$$
\frac{3}{14}(19.4x - 0.5) = 4.2x
$$

Multiply both sides by 14:
$$
3(19.4x - 0.5) = 58.8x
$$

Distribute:
$$
58.2x - 1.5 = 58.8x
$$

Subtract $ 58.2x $:
$$
-1.5 = 0.6x
$$

Divide by 0.6:
$$
x = \frac{-1.5}{0.6} = -2.5
$$

✔ Answer: $ x = -2.5 $

---

24) $ 0.541x + 0.81 = 0.416x + 1.56 $

Subtract $ 0.416x $ from both sides:
$$
0.125x + 0.81 = 1.56
$$

Subtract 0.81:
$$
0.125x = 0.75
$$

Divide by 0.125:
$$
x = \frac{0.75}{0.125} = 6
$$

✔ Answer: $ x = 6 $

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## ✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ x = 15 $ |
| 2 | $ x = 42.5 $ |
| 3 | $ x = 15.6 $ |
| 4 | $ x = 62.75 $ |
| 5 | $ x = 0.5 $ |
| 6 | $ x = 0.48 $ |
| 7 | $ x = 15,000 $ |
| 8 | $ x = 0.0006 $ |
| 9 | $ x = 2.1 $ |
| 10 | $ x = 14 $ |
| 11 | $ x = 120 $ |
| 12 | $ x = 70 $ |
| 13 | $ x = 0.15 $ |
| 14 | $ x = 0.39 $ |
| 15 | $ x = 0.525 $ |
| 16 | $ x = 3.3 $ |
| 17 | $ x = 0.3 $ |
| 18 | $ x = 0.4 $ |
| 19 | $ x = 0.2 $ |
| 20 | $ x = 0.075 $ |
| 21 | $ x = 6.25 $ |
| 22 | $ x = 50.25 $ |
| 23 | $ x = -2.5 $ |
| 24 | $ x = 6 $ |

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