1) The document contains solutions to various math problems presented without showing the work. 2) The problems include algebra, trigonometry, geometry, calculations with rates and time, and physics equations. 3) The solutions are presented with final answers only without showing the steps taken to arrive at each solution.

1. SOLUTION KEY TO MST
PRELIM
by Juan Apolinario C.
Reyes

2. −10 − 5 − 5 × 6 =
Solution
−10 − 5 − 30
= −15 − 30 = −45

3. Solution
3
4
+
1
2
−
2
3
=
12
=
9 + 6 − 8
12
=
7
12

4. Solution
−1 3
70 ÷ 10 − 14 = −7 − 14 = −21

5. 2 − 𝑥 2 + 𝑥 + 4 + 2𝑥2
= 4 + 2𝑥 − 2𝑥 − 𝑥2
+ 4 + 2𝑥2
Solution
= 8 + 𝑥2

6. Solution
2
3
−
4
3
=
2 − 4
3
= −
2
3
−
2
3
1
3
= −
2
3
× 3 = −2

7. Solution
5𝑥𝑦
𝑤𝑧
×
−5𝑥4
𝑤3
𝑦2
= −
25𝑥4+1
𝑦1−2
𝑤3−1
𝑧
= −
25𝑥5
𝑦−1
𝑤2
𝑧
= −
25𝑥5
𝑤2
𝑧𝑦

8. Solution
45 − 2𝑥 = 5𝑥 − 4
−5𝑥 − 2𝑥 = −4 − 45
−7𝑥 = −49
𝑥 =
49
7
= 7

9. 2
5
𝑥 − 3𝑥 + 10 = 2𝑥 − 3
5
2
5
𝑥 − 3𝑥 + 10 = 2𝑥 − 3
Solution
2𝑥 − 15𝑥 + 50 = 10𝑥 − 15
2𝑥 − 15𝑥 − 10𝑥 = −15 − 50
−23𝑥 = −65
𝑥 =
65
23
= 2
19
23

10. Solution
6
3 − 2𝑥
= 7
3 − 2𝑥
6
=
1
7
3 − 2𝑥 =
6
7
−2𝑥 =
6
7
− 3 =
6 − 21
7
= −
15
7
𝑥 =
15
7
∙
1
2
=
15
14
6
3 − 2𝑥
6
=
1
7

11. Solution
3 − 𝑥 2𝑥 + 5 = 0
3 − 𝑥 = 0 or 2𝑥 + 5 = 0
𝑥 = 3 or 2𝑥 = −5 → 𝑥 = −
5
2
Product Rule
If AB = 0,
then
A = 0 or B =
0

12. Solution
6𝑥 − 5 2
+ 10 = 10
6𝑥 − 5 2
= 0
6𝑥 − 5 = 0
𝑥 =
5
6
6𝑥 = 5

13. 𝑥2
− 18𝑥 + 81 = 0
𝑥 − 9 2
= 0
Solution
𝑥 = 9
𝑥 − 9 2
= 𝑥 − 9 𝑥 − 9
= 𝑥2
− 9𝑥 − 9𝑥 + 81
= 𝑥2
− 18𝑥 + 81

14. Solution
𝑝𝑥 = 𝑇
𝑝200 = 15000
𝑝 =
15000
200
=
150
2
= 75 php

15. Solution
𝑃𝐴 = 𝑝
1000 × 𝐴 = 1000 × 15 × 20 = 𝑝
1000 × 300 = 300000 php

16. Solution
𝑥2
− 8𝑦 − 9𝑥 = 22
− 8 −1 − 9 2
4 + 8 − 18 = −6

17. Solution
𝑥 𝑚, 𝑦 𝑚
𝑥 𝑚 =
5 + (−5)
2
= 0
𝑦 𝑚 =
7 + 11
2
= 9
𝑥 𝑚, 𝑦 𝑚 = 0, 9

18. Solution
𝑚 =
∆𝑦
∆𝑥
𝑚 =
11 − 4
3 − −4
=
7
7
= 1

19. Solution
𝑑 = 𝑥2 − 𝑥1
2 + 𝑦2 − 𝑦1
2
𝑑 = −7 2 + −7 2
𝑑 = 2 × 49 = 7 2

20. Solution
𝑐 = 𝑎2 + 𝑏2 = 64 + 25 𝑐 = 89 m

21. Solution
180° = 90° + 50°35′
+ 𝛼
𝛼 = 90° − 50°35′
𝛼 = 89°60′
− 50°35′
𝛼 = 39°25′

22. 𝐴 =
1
2
𝑏ℎ = 20 m2

23. Solution
𝜋𝑟2
= 36𝜋 𝑟2
=36
𝑟 = 6 m

24. Solution
𝐶 = 2𝜋𝑟
𝐶 = 12𝜋 m

25. Solution
tan 𝛽 =
5
10
tan 𝛽 =
1
2

26. Solution
cot 𝛼 =
5
10
=
1
2
tan 𝛼 =
10
5
= 2

27. Solution
cos 𝛽 =
𝐴
𝐻
𝑐 = 102 + 52 = 125 = 25 × 5
𝑐 = 5 5
cos 𝛽 =
10
5 5
=
2
5
cos 𝛽 =
2
5
∙
5
5
cos 𝛽 =
2 5
5

28. Solution
180° = 25° + 75° + 𝛾 𝛾 = 180° − 25° − 75° = 80°

29. 𝑎
sin 𝛼
=
𝑏
sin 𝛽
=
𝑐
sin 𝛾
𝑎
sin 25°
=
10
sin 75°

30. Solution
𝑑 = 𝑠 × 𝑡
𝑑 = 37 × 3 = 111 km

31. 111 km
36 km
𝑐 = 1112 + 362 km