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Solving equations is fundamental for Human Resource professionals who analyze workforce data, forecast staffing needs, or calculate compensation models. Mastery of linear and quadratic equations enables HR managers to interpret statistical models and optimize resource allocation effectively. This chapter focuses on solving linear equations with one or more variables and quadratic equations using the formula and factorization methods, providing practical applications relevant to HR contexts in Kenya.
Linear equations appear frequently in HR tasks such as budgeting salaries, calculating leave balances, or determining training costs. These equations involve variables raised only to the first power and can have one or multiple variables requiring simultaneous solutions.
A linear equation with one variable has the form \( ax + b = 0 \), where \( a \) and \( b \) are constants and \( x \) is the variable. The goal is to isolate \( x \) on one side.
$$ ax + b = 0 $$
Example 1: An HR officer calculates that the total monthly training cost is Ksh 10,000 plus Ksh 500 per employee. If the total budget is Ksh 35,000, find the number of employees \( x \) that can be trained.
Given: \( 500x + 10,000 = 35,000 \)
$$ 500x + 10,000 = 35,000 $$
$$ 500x = 35,000 - 10,000 $$
$$ 500x = 25,000 $$
$$ x = \frac{25,000}{500} $$
Answer: 50 employees
Example 2: A monthly salary after tax is Ksh 72,000, and the tax rate is 20%. Express the salary before tax as \( x \) and find its value.
Given: \( 0.8x = 72,000 \)
$$ 0.8x = 72,000 $$
$$ x = \frac{72,000}{0.8} $$
$$ x = 90,000 $$
Answer: Ksh 90,000
Example 3: An HR payroll system deducts Ksh 1,200 for pension contributions from a fixed salary \( x \). If the net salary is Ksh 48,800, find the gross salary.
Given: \( x - 1,200 = 48,800 \)
$$ x - 1,200 = 48,800 $$
$$ x = 48,800 + 1,200 $$
$$ x = 50,000 $$
Answer: Ksh 50,000
When HR managers analyze relationships involving two variables, such as overtime hours and pay, simultaneous linear equations are used. These take the form:
$$ a_1x + b_1y = c_1 $$
$$ a_2x + b_2y = c_2 $$
where \( x \) and \( y \) are variables.
Example 1: An HR officer calculates that 3 hours of overtime and 2 hours of training cost Ksh 6,000, while 5 hours of overtime and 1 hour of training cost Ksh 9,000. Find the cost per hour of overtime \( x \) and training \( y \).
Given:
$$ 3x + 2y = 6,000 $$
$$ 5x + y = 9,000 $$
Multiply the second equation by 2:
$$ 10x + 2y = 18,000 $$
Subtract the first equation:
$$ (10x + 2y) - (3x + 2y) = 18,000 - 6,000 $$
$$ 7x = 12,000 $$
$$ x = \frac{12,000}{7} = 1,714.29 $$
Substitute \( x \) into the second equation:
$$ 5(1,714.29) + y = 9,000 $$
$$ 8,571.45 + y = 9,000 $$
$$ y = 9,000 - 8,571.45 = 428.55 $$
Answer: Overtime cost = Ksh 1,714.29/hour, Training cost = Ksh 428.55/hour
Example 2: In a SACCO, the total contribution from 50 employees is Ksh 150,000. Regular members contribute \( x \) Ksh each and senior members contribute \( y \) Ksh each. If there are 30 regular and 20 senior members, and total contributions are Ksh 150,000, find \( x \) and \( y \) given that senior members contribute Ksh 1,000 more each than regular members.
Given:
$$ 30x + 20y = 150,000 $$
$$ y = x + 1,000 $$
Substitute second into first:
$$ 30x + 20(x + 1,000) = 150,000 $$
$$ 30x + 20x + 20,000 = 150,000 $$
$$ 50x = 130,000 $$
$$ x = \frac{130,000}{50} = 2,600 $$
Find \( y \):
$$ y = 2,600 + 1,000 = 3,600 $$
Answer: Regular contribution = Ksh 2,600, Senior contribution = Ksh 3,600
Example 3: A hotel pays Ksh 300 per hour for regular staff and Ksh 450 for supervisors. If total wage for 40 hours is Ksh 15,000 and total hours worked is 40, find the number of regular staff hours \( x \) and supervisor hours \( y \).
Given:
$$ 300x + 450y = 15,000 $$
$$ x + y = 40 $$
Express \( y = 40 - x \):
$$ 300x + 450(40 - x) = 15,000 $$
$$ 300x + 18,000 - 450x = 15,000 $$
$$ -150x = -3,000 $$
$$ x = \frac{3,000}{150} = 20 $$
Find \( y \):
$$ y = 40 - 20 = 20 $$
Answer: Regular staff worked 20 hours, Supervisors worked 20 hours
Complex HR models may require solving systems with three variables, such as calculating costs involving salaries, benefits, and bonuses. The system is:
$$ a_1x + b_1y + c_1z = d_1 $$
$$ a_2x + b_2y + c_2z = d_2 $$
$$ a_3x + b_3y + c_3z = d_3 $$
Example 1: A county government allocates funds for salaries (\( x \)), benefits (\( y \)), and training (\( z \)) totaling Ksh 1,200,000. The following conditions apply:
$$ x + y + z = 1,200,000 $$
$$ 2x + 3y + z = 2,500,000 $$
$$ x + 2y + 4z = 2,000,000 $$
Find \( x, y, z \).
Step 1: From the first equation express \( z = 1,200,000 - x, y \).
Step 2: Substitute into second and third:
Second:
$$ 2x + 3y + (1,200,000 - x, y) = 2,500,000 $$
$$ (2x, x) + (3y, y) + 1,200,000 = 2,500,000 $$
$$ x + 2y = 1,300,000 $$
Third:
$$ x + 2y + 4(1,200,000 - x, y) = 2,000,000 $$
$$ x + 2y + 4,800,000 - 4x - 4y = 2,000,000 $$
$$ -3x - 2y = -2,800,000 $$
Multiply the second simplified equation by 2:
$$ 2x + 4y = 2,600,000 $$
Add to third:
$$ (-3x - 2y) + (2x + 4y) = -2,800,000 + 2,600,000 $$
$$ - x + 2y = -200,000 $$
From earlier:
$$ x + 2y = 1,300,000 $$
Add both:
$$ (x + 2y) + (-x + 2y) = 1,300,000 - 200,000 $$
$$ 4y = 1,100,000 $$
$$ y = 275,000 $$
Find \( x \):
$$ x + 2(275,000) = 1,300,000 $$
$$ x + 550,000 = 1,300,000 $$
$$ x = 750,000 $$
Find \( z \):
$$ z = 1,200,000 - 750,000 - 275,000 = 175,000 $$
Answer: Salary = Ksh 750,000, Benefits = Ksh 275,000, Training = Ksh 175,000
Example 2: A retail business pays fixed monthly salaries \( x \), commissions \( y \), and bonuses \( z \) to sales staff with the following:
$$ x + y + z = 500,000 $$
$$ 2x, y + 3z = 1,000,000 $$
$$ 4x + y, z = 1,200,000 $$
Find \( x, y, z \).
Step 1: From first:
$$ z = 500,000 - x, y $$
Step 2: Substitute into second and third:
Second:
$$ 2x, y + 3(500,000 - x, y) = 1,000,000 $$
$$ 2x, y + 1,500,000 - 3x - 3y = 1,000,000 $$
$$ - x - 4y = -500,000 $$
Third:
$$ 4x + y - (500,000 - x, y) = 1,200,000 $$
$$ 4x + y - 500,000 + x + y = 1,200,000 $$
$$ 5x + 2y = 1,700,000 $$
Step 3: Multiply second equation by 2:
$$ -2x - 8y = -1,000,000 $$
Add to third:
$$ (5x + 2y) + (-2x - 8y) = 1,700,000 - 1,000,000 $$
$$ 3x - 6y = 700,000 $$
Step 4: Express \( x \):
$$ 3x = 700,000 + 6y $$
$$ x = \frac{700,000 + 6y}{3} $$
Step 5: Substitute into second simplified:
$$ - \frac{700,000 + 6y}{3} - 4y = -500,000 $$
Multiply all by 3:
$$ -(700,000 + 6y) - 12y = -1,500,000 $$
$$ -700,000 - 6y - 12y = -1,500,000 $$
$$ -700,000 - 18y = -1,500,000 $$
$$ -18y = -800,000 $$
$$ y = \frac{800,000}{18} = 44,444.44 $$
Step 6: Find \( x \):
$$ x = \frac{700,000 + 6(44,444.44)}{3} = \frac{700,000 + 266,666.64}{3} = \frac{966,666.64}{3} = 322,222.21 $$
Step 7: Find \( z \):
$$ z = 500,000 - 322,222.21 - 44,444.44 = 133,333.35 $$
Answer: Salary = Ksh 322,222.21, Commission = Ksh 44,444.44, Bonus = Ksh 133,333.35
Two key methods for solving simultaneous linear equations are substitution and elimination. Both are useful for HR professionals depending on the problem structure.
Example 1 (Substitution): An HR manager finds that the number of permanent employees \( x \) and contract employees \( y \) satisfy:
$$ x + y = 80 $$
$$ 2x + 3y = 200 $$
Use substitution to find \( x \) and \( y \).
From first:
$$ y = 80 - x $$
Substitute into second:
$$ 2x + 3(80 - x) = 200 $$
$$ 2x + 240 - 3x = 200 $$
$$ - x = -40 $$
$$ x = 40 $$
Find \( y \):
$$ y = 80 - 40 = 40 $$
Answer: Permanent = 40, Contract = 40
Example 2 (Elimination): Given:
$$ 3x + 2y = 18 $$
$$ 5x - 2y = 22 $$
Use elimination to find \( x \) and \( y \).
Add equations:
$$ (3x + 2y) + (5x - 2y) = 18 + 22 $$
$$ 8x = 40 $$
$$ x = 5 $$
Substitute \( x \) in first:
$$ 3(5) + 2y = 18 $$
$$ 15 + 2y = 18 $$
$$ 2y = 3 $$
$$ y = 1.5 $$
Answer: \( x = 5 \), \( y = 1.5 \)
An HR department budgets Ksh 12,000 fixed cost and Ksh 800 per trainee. If the total budget is Ksh 44,000, find the number of trainees. (3 marks)
In a training program, 4 hours of theory and 3 hours of practical cost Ksh 11,000, while 2 hours of theory and 5 hours of practical cost Ksh 9,000. Find the cost per hour of theory and practical. (5 marks)
A company pays Ksh 250 per hour for junior staff, Ksh 400 per hour for senior staff, and Ksh 600 per hour for supervisors. For 100 total hours, the total wage is Ksh 40,000, and the wage for senior staff and supervisors combined is Ksh 28,000. Find the hours worked by each category. (7 marks)
Solve by elimination: \( 4x + 3y = 27 \) and \( 5x - 3y = 18 \). (4 marks)
Using substitution, solve \( x + 2y = 10 \) and \( 3x, y = 5 \). (4 marks)
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Create a free accountThis chapter covered the application of statistical equations starting with solving linear equations involving one or more variables, providing methods to find unknown values in business contexts. It then explored quadratic equations, focusing on the formula method and factorization as techniques to determine variable roots. The chapter progressed to solving simultaneous equations using substitution, elimination, and matrix methods, enabling the resolution of multiple interrelated equations efficiently. Breakeven analysis was computed to identify the point where total revenue equals total cost, a critical concept in business decision-making. Calculus was introduced through differentiation and integration, tools essential for analyzing rates of change and cumulative quantities in economic models. Finally, the formulation of total revenue, total cost, and profit equations was addressed, linking mathematical expressions to practical business performance evaluation. Together, these topics equip students with quantitative skills to analyze and solve complex business problems.
A Human Resource manager needs to find the monthly salary \(x\) if the total remuneration package is given by the equation \(2x + 5000 = 25000\). Find \(x\). (2 marks)
Solve the quadratic equation \(x^2 - 5x + 6 = 0\) using the formula method to find the possible values of \(x\). (3 marks)
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