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Systems of Linear Equations

Grade 10 · Mathematics · Worksheet 2

  1. Sophia is managing the production of two types of custom furniture: a basic chair and an armchair. Each basic chair requires 7 units of wood and 4 units of upholstery fabric. Each armchair requires 9 units of wood and 8 units of upholstery fabric. The workshop has exactly 189 units of wood and 124 units of upholstery fabric available for the week. If Sophia wants to use all available materials exactly, how many of each type of chair should be produced? Answer: ______________
  2. Matiu is managing the inventory for a community garden that sells two types of compost: a standard mix and a premium mix. Each bag of standard mix requires 5 kilograms of green waste and 3 kilograms of brown waste. Each bag of premium mix requires 7 kilograms of green waste and 5 kilograms of brown waste. The garden has exactly 185 kilograms of green waste and 115 kilograms of brown waste available. Matiu wants to use all the green waste and all the brown waste exactly, with no leftovers. How many bags of each type of compost should he produce? Answer: ______________
  3. Emma is organizing a school fundraiser selling two types of gift baskets: a standard basket and a deluxe basket. Each standard basket requires 3 hours of preparation and 1 hour of packaging, while each deluxe basket requires 5 hours of preparation and 3 hours of packaging. The volunteer team has exactly 51 hours available for preparation and 25 hours available for packaging each week. If Emma wants to use all available hours exactly, how many of each type of gift basket should they produce each week? Answer: ______________
  4. A technology company is analyzing the profitability of two different product lines. Product A has a fixed monthly cost of $800 and generates $25 profit per unit sold. Product B has a fixed monthly cost of $1200 and generates $40 profit per unit sold. The company wants to determine how many units of each product must be sold for both product lines to have the same total monthly profit. Write your answer as an ordered pair (units of A, units of B). Answer: ______________
  5. A technology company is analyzing the relationship between advertising spending and product sales. They found that when they spend $x thousand on online ads and $y thousand on TV ads, their monthly revenue follows the system: R₁ = 2x + 3y + 10 and R₂ = 4x + y + 5, where both equations must be equal for optimal budget allocation. If the company wants to achieve revenue balance between these two advertising strategies, find the combination of online and TV advertising spending (in thousands) that satisfies this condition. Answer: ______________
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Answer Key & Explanations

Systems of Linear Equations · Grade 10 · Worksheet 2

  1. Sophia is managing the production of two types of custom furniture: a basic chair and an armchair. Each basic chair requires 7 units of wood and 4 units of upholstery fabric. Each armchair requires 9 units of wood and 8 units of upholstery fabric. The workshop has exactly 189 units of wood and 124 units of upholstery fabric available for the week. If Sophia wants to use all available materials exactly, how many of each type of chair should be produced? Answer: Basic chairs: 15, Armchairs: 8 Solution: Let x = number of basic chairs, y = number of armchairs. Wood equation: 7x + 9y = 189 Fabric equation: 4x + 8y = 124 Solve using elimination.
    Full step-by-step solution

    Step 1: Let x = number of basic chairs, y = number of armchairs. Step 2: Wood equation: 7x + 9y = 189 Step 3: Fabric equation: 4x + 8y = 124 Step 4: Solve using elimination. Multiply the second equation by 7 to align x coefficients: 28x + 56y = 868 Step 5: Multiply the first equation by 4: 28x + 36y = 756 Step 6: Subtract the second equation from the first: (28x + 36y) - (28x + 56y) = 756 - 868 Step 7: Simplify: -20y = -112, so y = 8 Step 8: Substitute y = 8 into 7x + 9(8) = 189: 7x + 72 = 189, so 7x = 117, thus x = 15 Step 9: Check with fabric: 4(15) + 8(8) = 60 + 64 = 124. Correct. The answer is 15 basic chairs and 8 armchairs.

  2. Matiu is managing the inventory for a community garden that sells two types of compost: a standard mix and a premium mix. Each bag of standard mix requires 5 kilograms of green waste and 3 kilograms of brown waste. Each bag of premium mix requires 7 kilograms of green waste and 5 kilograms of brown waste. The garden has exactly 185 kilograms of green waste and 115 kilograms of brown waste available. Matiu wants to use all the green waste and all the brown waste exactly, with no leftovers. How many bags of each type of compost should he produce? Answer: Standard: 20, Premium: 15 Solution: Let x = number of standard bags, y = number of premium bags. Green waste equation: 5x + 7y = 185. Brown waste equation: 3x + 5y = 115.
    Full step-by-step solution

    Step 1: Let x = number of standard bags, y = number of premium bags. Step 2: Green waste equation: 5x + 7y = 185. Step 3: Brown waste equation: 3x + 5y = 115. Step 4: Solve using elimination. Multiply the green waste equation by 3 and the brown waste equation by 5 to align x coefficients: 3(5x + 7y) = 3(185) → 15x + 21y = 555 5(3x + 5y) = 5(115) → 15x + 25y = 575 Step 5: Subtract the first new equation from the second: (15x + 25y) - (15x + 21y) = 575 - 555 4y = 20 y = 5 Step 6: Substitute y = 5 into the original green waste equation: 5x + 7(5) = 185 5x + 35 = 185 5x = 150 x = 30 Step 7: Check with brown waste: 3(30) + 5(5) = 90 + 25 = 115. Correct. The answer is 30 standard bags and 5 premium bags.

  3. Emma is organizing a school fundraiser selling two types of gift baskets: a standard basket and a deluxe basket. Each standard basket requires 3 hours of preparation and 1 hour of packaging, while each deluxe basket requires 5 hours of preparation and 3 hours of packaging. The volunteer team has exactly 51 hours available for preparation and 25 hours available for packaging each week. If Emma wants to use all available hours exactly, how many of each type of gift basket should they produce each week? Answer: (13, 4) Solution: Let x = number of standard baskets, y = number of deluxe baskets. Write the preparation hours equation: 3x + 5y = 51. Write the packaging hours equation: x + 3y = 25.
    Full step-by-step solution

    Step 1: Let x = number of standard baskets, y = number of deluxe baskets. Step 2: Write the preparation hours equation: 3x + 5y = 51. Step 3: Write the packaging hours equation: x + 3y = 25. Step 4: Solve using elimination. Multiply the second equation by 3: 3x + 9y = 75. Step 5: Subtract the first equation from this result: (3x + 9y) - (3x + 5y) = 75 - 51. Step 6: Simplify: 4y = 24, so y = 6. Step 7: Substitute y = 6 into x + 3y = 25: x + 3(6) = 25, so x + 18 = 25, thus x = 7. Step 8: Check: Preparation: 3(7) + 5(6) = 21 + 30 = 51 hours. Packaging: 7 + 3(6) = 7 + 18 = 25 hours. The solution is (7, 6). Emma should produce 7 standard baskets and 6 deluxe baskets each week.

  4. A technology company is analyzing the profitability of two different product lines. Product A has a fixed monthly cost of $800 and generates $25 profit per unit sold. Product B has a fixed monthly cost of $1200 and generates $40 profit per unit sold. The company wants to determine how many units of each product must be sold for both product lines to have the same total monthly profit. Write your answer as an ordered pair (units of A, units of B). Answer: (16, 10) Solution: In business applications, systems of equations can model scenarios where different options have the same outcome.
    Full step-by-step solution

    In business applications, systems of equations can model scenarios where different options have the same outcome. The key is to define variables for the unknown quantities, write expressions for total profit that include both fixed costs and variable profits, then set the expressions equal to find when the outcomes match. This approach helps compare different business strategies mathematically.

  5. A technology company is analyzing the relationship between advertising spending and product sales. They found that when they spend $x thousand on online ads and $y thousand on TV ads, their monthly revenue follows the system: R₁ = 2x + 3y + 10 and R₂ = 4x + y + 5, where both equations must be equal for optimal budget allocation. If the company wants to achieve revenue balance between these two advertising strategies, find the combination of online and TV advertising spending (in thousands) that satisfies this condition. Answer: x = 1, y = 3 Solution: In business applications, setting two different revenue models equal to each other helps find the point where both strategies are equally effective.
    Full step-by-step solution

    In business applications, setting two different revenue models equal to each other helps find the point where both strategies are equally effective. This creates a linear equation that can be solved for one variable in terms of the other, or you can use substitution to find specific values that satisfy the condition.