Algebra 2 Final Review - Step-by-Step
for Difficult Problems
Solving a System of 3 Equations with 3 Variables
Problem:
x + y + z = 10
3x − y − z = 22
2x − 4y − 3z = −13
Steps:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Step 1: Use Eq1 and Eq2 to eliminate z.
Add Eq1 and Eq2: (x + y + z) + (3x − y − z) = 10 + 22 → 4x = 32 → x = 8
Step 2: Substitute x = 8 into Eq1 and Eq3 to eliminate x.
From Eq1: 8 + y + z = 10 → y + z = 2 (Eq4)
From Eq3: 2(8) − 4y − 3z = −13 → 16 − 4y − 3z = −13 → −4y − 3z = −29 (Eq5)
Step 3: Solve Eq4 and Eq5.
From Eq4: y = 2 − z. Substitute into Eq5: −4(2 − z) − 3z = −29
→ −8 + 4z − 3z = −29 → z = −21, then y = 2 − (−21) = 23
Final answer: x = 8, y = 23, z = −21
Matrix Multiplication (AB vs. BA)
Problem:
A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]
Steps:
10. Step 1: Multiply A × B.
11. First row, first column: (1×5 + 2×7) = 5 + 14 = 19
12. First row, second column: (1×6 + 2×8) = 6 + 16 = 22
13. Second row, first column: (3×5 + 4×7) = 15 + 28 = 43
14. Second row, second column: (3×6 + 4×8) = 18 + 32 = 50
15. A × B = [[19, 22], [43, 50]]
16. Step 2: Multiply B × A to confirm it's not the same.
17. First row, first column: (5×1 + 6×3) = 5 + 18 = 23
18. B × A = [[23, 34], [31, 46]] ≠ A × B → Not commutative
Using Cramer’s Rule
Problem:
2x − y = 1
3x + 3y = 24
Steps:
19. Step 1: Coefficient matrix: |2 -1|
|3 3|
20. Determinant D = (2)(3) − (−1)(3) = 6 + 3 = 9
21. Dx: Replace x-column with constants: |1 -1|
|24 3| → Dx = (1)(3) − (−1)(24) = 3 + 24 = 27
22. Dy: Replace y-column with constants: |2 1|
|3 24| → Dy = (2)(24) − (1)(3) = 48 − 3 = 45
23. x = Dx/D = 27/9 = 3, y = Dy/D = 45/9 = 5
Factoring a Complex Trinomial
Problem:
10x² − 27x + 52
Steps:
24. Step 1: Multiply a*c = 10 * 52 = 520
25. Step 2: Find two numbers that multiply to 520 and add to −27 → None found.
26. Conclusion: The trinomial may be prime or require completing the square/quadratic
formula.
Scientific Notation with Operations
Problem:
2.8 × 10⁻¹³ × 4 × 10⁹
Steps:
27. Step 1: Multiply the numbers: 2.8 × 4 = 11.2
28. Step 2: Add exponents: −13 + 9 = −4
29. Answer: 11.2 × 10⁻⁴ = 1.12 × 10⁻³ (in proper scientific notation)