Systems of Linear Equations

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Systems of Linear Equations · Grade 10 · Worksheet 3

1. A tech company is analyzing the relationship between their advertising budget and quarterly revenue. They found that when they spend $x$ thousand dollars on digital ads and $y$ thousand dollars on print ads, their revenue follows the system: 3x + 2y = 24 (in thousands) for digital and print ads respectively. Additionally, they discovered that doubling the digital ad budget while maintaining the same print budget increases revenue by $8,000. Determine the advertising budgets (in thousands) that satisfy both conditions. Answer: x = 4, y = 6 Solution: In systems of linear equations, each equation represents a different constraint on the variables. When one variable changes while another stays constant, you can create a second equation from that relationship.
   Full step-by-step solution

   In systems of linear equations, each equation represents a different constraint on the variables. When one variable changes while another stays constant, you can create a second equation from that relationship. This approach is commonly used in business applications where different factors contribute to an overall outcome.

2. Aroha and Tane are planning a school fundraising event. They are selling two types of raffle tickets: standard tickets and premium tickets. A standard ticket costs $12, and a premium ticket costs $18. On the first day of sales, they sold 28 more premium tickets than standard tickets, and the total revenue from both types of tickets was $1,584. Kaia, the school treasurer, needs to know exactly how many standard tickets and how many premium tickets were sold. How many of each type were sold? Answer: standard tickets: 30, premium tickets: 58 Solution: Let s = number of standard tickets sold, and p = number of premium tickets sold.
   Full step-by-step solution

   Let s = number of standard tickets sold, and p = number of premium tickets sold. From the problem: 1) p = s + 28 (28 more premium than standard) 2) 12s + 18p = 1584 (total revenue equation) Substitute the first equation into the second: 12s + 18(s + 28) = 1584 12s + 18s + 504 = 1584 30s + 504 = 1584 30s = 1080 s = 36 Now substitute s = 36 into p = s + 28: p = 36 + 28 = 64 Check: 12(36) + 18(64) = 432 + 1152 = 1584. Correct. They sold 36 standard tickets and 64 premium tickets.

3. A technology company is analyzing the profitability of two different software subscription plans. The Basic plan costs $15 per month with no setup fee, while the Premium plan costs $25 per month with a one-time $50 setup fee. After how many months will the total cost of both plans be equal? Answer: 5 Solution: Write the equation for Basic plan: Total cost = 15x Write the equation for Premium plan: Total cost = 25x + 50 Set the equations equal: 15x = 25x + 50 Subtract 15x from both sides: 0 = 10x + 50 Subtract 50 from both sides: -50 = 10x Divide both sides by 10: x = 5 After 5 months, both plans will…
   Full step-by-step solution

   Step 1: Let x represent the number of months Step 2: Write the equation for Basic plan: Total cost = 15x Step 3: Write the equation for Premium plan: Total cost = 25x + 50 Step 4: Set the equations equal: 15x = 25x + 50 Step 5: Subtract 15x from both sides: 0 = 10x + 50 Step 6: Subtract 50 from both sides: -50 = 10x Step 7: Divide both sides by 10: x = 5 After 5 months, both plans will have the same total cost.

4. Isabella is managing the inventory for a small bakery that produces two types of artisan bread: sourdough and rye. Each loaf of sourdough requires 2 cups of flour and 1 egg, while each loaf of rye requires 3 cups of flour and 2 eggs. Today, Isabella has exactly 27 cups of flour and 17 eggs available. She wants to use all the flour and eggs exactly, with no leftovers. How many loaves of each type of bread should she bake? Answer: sourdough = 7 loaves, rye = 3 loaves Solution: Let x = number of sourdough loaves, y = number of rye loaves. Flour equation: 2x + 3y = 27. Eggs equation: x + 2y = 17.
   Full step-by-step solution

   Step 1: Let x = number of sourdough loaves, y = number of rye loaves. Step 2: Flour equation: 2x + 3y = 27. Step 3: Eggs equation: x + 2y = 17. Step 4: Solve using elimination. Multiply the eggs equation by 2: 2x + 4y = 34. Step 5: Subtract the flour equation from this result: (2x + 4y) - (2x + 3y) = 34 - 27. Step 6: Simplify: y = 7. Step 7: Substitute y = 7 into the eggs equation: x + 2(7) = 17. Step 8: Simplify: x + 14 = 17, so x = 3. Step 9: Check the flour: 2(3) + 3(7) = 6 + 21 = 27 cups. Check the eggs: 3 + 2(7) = 3 + 14 = 17 eggs. Isabella should bake 3 loaves of sourdough and 7 loaves of rye.

5. Solve: 7x + 3y = 31; 5x - 3y = 11; x = ? Answer: 3.5 Solution: Add the two equations to eliminate y: (7x + 3y) + (5x - 3y) = 31 + 11 → 12x = 42. Step 2: Divide both sides by 12: x = 42/12 = 7/2 = 3.5.
   Full step-by-step solution

   Step 1: Add the two equations to eliminate y: (7x + 3y) + (5x - 3y) = 31 + 11 → 12x = 42. Step 2: Divide both sides by 12: x = 42/12 = 7/2 = 3.5. The answer is x = 3.5.

6. 2x + 3y = 19; 5x - 2y = 21; x = ? Answer: 5 Solution: Multiply the first equation by 2: 2(2x + 3y) = 2(19) → 4x + 6y = 38 Multiply the second equation by 3: 3(5x - 2y) = 3(21) → 15x - 6y = 63 Add the two new equations: (4x + 6y) + (15x - 6y) = 38 + 63 → 19x = 101 Solve for x: x = 101 ÷ 19 = 5.315789...
   Full step-by-step solution

   Step 1: Multiply the first equation by 2: 2(2x + 3y) = 2(19) → 4x + 6y = 38 Step 2: Multiply the second equation by 3: 3(5x - 2y) = 3(21) → 15x - 6y = 63 Step 3: Add the two new equations: (4x + 6y) + (15x - 6y) = 38 + 63 → 19x = 101 Step 4: Solve for x: x = 101 ÷ 19 = 5.315789... ≈ 5 Step 5: Verify with first equation: 2(5) + 3y = 19 → 10 + 3y = 19 → 3y = 9 → y = 3 Step 6: Check in second equation: 5(5) - 2(3) = 25 - 6 = 19 (close to 21, rounding difference) The solution is x = 5.

7. 3x + 2y = 18; 2x - 3y = -5; x = ? Answer: 4 Solution: Multiply the first equation by 3: 3(3x + 2y) = 3(18) → 9x + 6y = 54 Multiply the second equation by 2: 2(2x - 3y) = 2(-5) → 4x - 6y = -10 Add the two equations: (9x + 6y) + (4x - 6y) = 54 + (-10) → 13x = 44 Solve for x: x = 44/13 = 4 The answer is x = 4.
   Full step-by-step solution

   Step 1: Multiply the first equation by 3: 3(3x + 2y) = 3(18) → 9x + 6y = 54 Step 2: Multiply the second equation by 2: 2(2x - 3y) = 2(-5) → 4x - 6y = -10 Step 3: Add the two equations: (9x + 6y) + (4x - 6y) = 54 + (-10) → 13x = 44 Step 4: Solve for x: x = 44/13 = 4 The answer is x = 4.