🧠 Algebra

Summary Cheat Sheet

• Linear equations model first-degree relationships such as cost and break-even.
• Quadratic equations use discriminant to identify nature of roots.
• Simultaneous equations solve multi-input agricultural planning problems.

References

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This lesson explains core algebra tools used to model relationships in crop production, input planning, and farm economics.

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Introduction to Algebra

Algebra is the branch of mathematics that uses letters and symbols to represent unknown quantities and express relationships between variables. In agriculture, algebraic equations model relationships such as crop yield vs. fertilizer input, cost vs. production, and growth rates. The ability to solve equations is fundamental to agricultural economics, experiment design, and resource optimization.

Linear Equations

A linear equation is an equation of the first degree (the highest power of the variable is 1). The general form is ax + b = 0, where a and b are constants and x is the variable.

Solving linear equations: Isolate the variable by performing inverse operations.

Example: A farmer spends Rs. 500 on seeds and Rs. 150 per bag of fertilizer. If his total expenditure is Rs. 2,300, how many bags of fertilizer did he buy?

• Equation: 500 + 150x = 2300
• 150x = 2300 - 500 = 1800
• x = 1800 / 150 = 12 bags

Linear equations also appear in break-even analysis: if fixed costs are Rs. 10,000 and variable cost per unit is Rs. 50 while selling price per unit is Rs. 80, the break-even point is: 10000 + 50x = 80x, giving x = 10000/30 = 334 units.

Quadratic Equations

A quadratic equation has the general form ax^2 + bx + c = 0, where a is not zero. The solutions (roots) are found using the quadratic formula:

x = (-b +/- sqrt(b^2 - 4ac)) / 2a

The discriminant (D = b^2 - 4ac) determines the nature of roots:

• D > 0: Two distinct real roots
• D = 0: Two equal real roots (one repeated root)
• D < 0: No real roots (complex roots)

Example: A rectangular field has a perimeter of 60 m and area of 200 sq m. Find the dimensions.

• Let length = x, then width = 30 - x (since 2x + 2w = 60, so x + w = 30)
• Area: x(30 - x) = 200
• 30x - x^2 = 200
• x^2 - 30x + 200 = 0
• Using the quadratic formula: x = (30 +/- sqrt(900 - 800)) / 2 = (30 +/- 10) / 2
• x = 20 or x = 10
• Dimensions: 20 m x 10 m

Simultaneous Equations

Simultaneous equations (systems of equations) involve two or more equations with two or more unknowns, solved together. Common methods include substitution and elimination.

Elimination method example: A farmer buys 3 kg of urea and 2 kg of DAP for Rs. 70. Another combination of 5 kg urea and 4 kg DAP costs Rs. 130. Find the price per kg.

• Equation 1: 3x + 2y = 70
• Equation 2: 5x + 4y = 130
• Multiply Equation 1 by 2: 6x + 4y = 140
• Subtract Equation 2: 6x + 4y - 5x - 4y = 140 - 130
• x = 10 (urea costs Rs. 10/kg)
• Substitute back: 3(10) + 2y = 70, so 2y = 40, y = 20 (DAP costs Rs. 20/kg)

Substitution method: Solve one equation for one variable, then substitute into the other. Both methods yield identical results and are chosen based on convenience. Simultaneous equations are used extensively in agricultural economics for input-output analysis, diet formulation for livestock, and fertilizer recommendation calculations.

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