Linear Equations & Interpolation

7.1 Setting up a linear equation

Step 1: Multiply both sides by the denominator to clear fractions
Step 2: Subtract known terms to isolate the unknown
Step 3: Divide by the coefficient to solve for x

7.2 Target-rate problems

x = target rate × Total − Current

Example: 630 000 employed out of 900 000. Target 80%. x = 0.80 × 900 000 − 630 000 = 90 000 additional jobs needed.

7.3 When the denominator also changes

(Numerator + x) ÷ (Total + x) = target rate
x = (rate × Total − Numerator) ÷ (1 − rate)

Example: 280 000 PT trips out of 640 000. Target 45%. x = (288 000 − 280 000) ÷ 0.55 ≈ 14 545.

7.4 Two unknowns: substitution method

If A = k × B, let B = x, so A = k × x
Solve for x, then recover A = k × x

Example: two funds total €52 000. Fund A is 3× Fund B. Let B = x: 4x = 52 000 → x = 13 000. B = €13 000, A = €39 000.

7.5 Linear interpolation

Annual increment = (End value − Start value) ÷ (End year − Start year)
Value at year t  = Start value + (t − Start year) × increment

Example: renewable share 14% in 2018, 26% in 2024. Increment = 2 pp/year. Estimate for 2021 = 14 + 3×2 = 20%.

7.6 Reverse tax calculation

Step 1: compute max tax for each lower bracket
Step 2: remaining tax = total − lower bracket taxes
Step 3: income in top bracket = remaining tax ÷ top rate
Step 4: gross income = sum of all bracket slices

Example: brackets 0%/20%/40%. Total tax = €13 000. Bracket 2 max = €5 000. Remaining = €8 000. Bracket 3 income = €20 000. Gross = €55 000.