Algebra, Geometry, and Financial Mathematics Essentials

Posted on May 25, 2026 in Mathematics

Algebra, Linear Graphs & Simultaneous Equations

Ex 1 (Simplify): 3x – 5y + 2x + 8y = 5x + 3y
Ex 2 (Expand & Solve): 3(x-4) = 2x+5 → 3x-12 = 2x+5 → x = 17
Tips:
- Only add/subtract exact like terms (same variables).
- Expand brackets by multiplying the outside term by everything inside.
- Move the smaller variable first to keep values positive.

Ex 1 (Solve): -3x + 4 ≤ 10 → -3x ≤ 6 → x ≥ -2
Tips:
- Flip the sign (< to >) whenever you multiply or divide by a negative number.
- Solid dot (•) for ≤ and ≥. Open dot (°) for < and >.

Ex 1 (Gradient formula): Points (1, 3) and (3, 7) → m = (7-3) / (3-1) = 4/2 = 2
Abacus Calculator Fraction Tip: Input fractions using the [ab/c] key. Press 4 then [ab/c] then 2 then [=] to simplify fractions instantly.
Ex 2 (Sketch via Intercepts): For y = -2x + 4
- y-intercept (let x=0): y = 4 → (0,4)
- x-intercept (let y=0): 0 = -2x+4 → 2x = 4 → x = 2 → (2,0)
Tips:
- m = gradient (rise / run). c = y-intercept.
- Parallel lines have equal gradients (m1 = m2).
- Perpendicular lines have negative reciprocal gradients (m1 * m2 = -1). [1]

Ex 1 (Substitution): Solve y = 3x – 1 and 2x + y = 9
- Substitute 1st into 2nd: 2x + (3x-1) = 9 → 5x-1 = 9 → 5x = 10 → x = 2
- Find y: y = 3(2)-1 = 5 → Solution: (2,5)
Ex 2 (Graphical): Plot both lines on one grid. Find where they cross. The intersection point (x, y) is the solution.
Tips:
- Use substitution when one equation already starts with y = or x =.

Ex 1 (SSS Test): If side AB = DE, side BC = EF, and side AC = DF, then Triangle ABC is identical to Triangle DEF.
Tips:
- Use 4 tests: SSS (3 sides), SAS (2 sides + middle angle), AAS (2 angles + 1 side), RHS (Right angle + Hypotenuse + 1 side).

Ex 1 (Finding Side): Small triangle side = 4, matching big triangle side = 12.
- Scale factor (k) = 12 / 4 = 3.
- If another small side is 5, the large missing side is 5 * 3 = 15.
Tips:
- Scale Factor (k) = Image side / Original side.
- Test with AAA (all angles equal) or matching side ratios.

Ex 1 (Simplify): (2x^5 * 3x^-2) / x^2 = 6x^(5 + -2) / x^2 = 6x^3 / x^2 = 6x^(3 – 2) = 6x^1 = 6x
Ex 2 (Negative Fractional Power): To calculate (2/3)^-2 on Abacus SX-II:
- Type: ( 2 [ab/c] 3 ) [x&block;] (-) 2 [=]
- Result will show as an improper fraction or mixed number. Press [2nd] then [d/c] to switch formats.
Tips:
- Multiply same bases → Add powers. Divide same bases → Subtract powers.
- Power outside a bracket → Multiply powers.
- Anything to the power of 0 equals 1 (x^0 = 1).
- Always use the dedicated negative key [(-)] for negative indices, NOT the minus subtraction key [-]. [1, 2, 3]

Ex 1 (Evaluate): Find y = 3 * 2^x when x = 3
- Abacus Entry: Type 3 * 2 [x&block;] 3 [=] → Result is 24.
Tips:
- General formula: y = a * b^x. (a is the starting value or y-intercept).
- If b > 1, it is growth (curves up). If b < 1, it is decay (curves down).
- Use the right arrow key [▶] to exit out of the power exponent screen box before typing next steps. [1]

Ex 1 (Calculate Balance): Invest $5000 at 4% per year compound interest for 3 years.
- Abacus Entry: Type 5000 * ( 1 + 4 [ab/c] 100 ) [x&block;] 3 [=]
- Total Balance (A) = $5624.32
- Interest Earned = Final Balance – Starting Principal = $5624.32 – $5000 = $624.32
Tips:
- Formula: A = P * (1 + r/100)^n
- A is the total final balance, not just the interest. To find interest alone, do A – P. [1]