Systems of Linear Equations

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Systems of Linear Equations · Grade 10 · Worksheet 2

1. Olivia is a concert promoter organizing two types of VIP packages for an upcoming tour. The Gold package requires 3 hours of backstage setup and 1 hour of soundcheck coordination, while the Platinum package requires 5 hours of backstage setup and 3 hours of soundcheck coordination. Due to venue constraints, Olivia has exactly 63 hours available for backstage setup and 33 hours available for soundcheck coordination across all packages. If she wants to use every available hour, how many of each VIP package should she offer? Answer: (9, 8) Solution: Let x = number of Gold packages, y = number of Platinum packages. Write the equation for backstage setup hours: 3x + 5y = 63 Write the equation for soundcheck coordination hours: 1x + 3y = 33 Solve using elimination.
   Full step-by-step solution

   Step 1: Let x = number of Gold packages, y = number of Platinum packages. Step 2: Write the equation for backstage setup hours: 3x + 5y = 63 Step 3: Write the equation for soundcheck coordination hours: 1x + 3y = 33 Step 4: Solve using elimination. Multiply the second equation by 3: 3x + 9y = 99 Step 5: Subtract the first equation from this result: (3x + 9y) - (3x + 5y) = 99 - 63 Step 6: Simplify: 4y = 36, so y = 9 Step 7: Substitute y = 9 into the second equation: x + 3(9) = 33 Step 8: Simplify: x + 27 = 33, so x = 6 Step 9: Check in the first equation: 3(6) + 5(9) = 18 + 45 = 63 (correct) Step 10: Check in the second equation: 6 + 3(9) = 6 + 27 = 33 (correct) Olivia should offer 6 Gold packages and 9 Platinum packages.

2. Liam is managing a small business that produces two types of artisanal candles: lavender and sandalwood. The lavender candles sell for $12 each, and the sandalwood candles sell for $15 each. Last weekend, Liam sold 40 candles in total and collected $540 in revenue. How many lavender candles did Liam sell? Answer: 20 Solution: Let L = number of lavender candles sold. Let S = number of sandalwood candles sold. 1.
   Full step-by-step solution

   Let's define variables for the number of each type of candle. Let L = number of lavender candles sold. Let S = number of sandalwood candles sold. We know: 1. The total number of candles sold is 40. So: L + S = 40 2. The lavender candles sell for $12 each, sandalwood for $15 each, and total revenue is $540. So: 12L + 15S = 540 We now have a system of equations: Equation 1: L + S = 40 Equation 2: 12L + 15S = 540 Step 1: Solve for one variable from Equation 1. From L + S = 40, we get S = 40 - L. Step 2: Substitute S = 40 - L into Equation 2. 12L + 15(40 - L) = 540 Step 3: Simplify and solve for L. 12L + 600 - 15L = 540 (12L - 15L) + 600 = 540 -3L + 600 = 540 Step 4: Isolate the L term. -3L = 540 - 600 -3L = -60 Step 5: Solve for L. L = (-60) / (-3) L = 20 So, Liam sold 20 lavender candles. Step 6: Check the answer. If L = 20, then S = 40 - 20 = 20. Revenue = 12*20 + 15*20 = 240 + 300 = 540. This matches the given total revenue. Final answer: 20

3. Liam is managing a small business that produces two types of handmade candles: lavender and sandalwood. The lavender candles require 2 ounces of wax and 0.5 ounces of fragrance oil each, while the sandalwood candles require 3 ounces of wax and 0.25 ounces of fragrance oil each. If Liam has 48 ounces of wax and 9 ounces of fragrance oil available, how many of each type of candle should he make to use all available supplies? Answer: lavender: 12, sandalwood: 8 Solution: This type of problem involves setting up a system of linear equations to model resource allocation.
   Full step-by-step solution

   This type of problem involves setting up a system of linear equations to model resource allocation. In manufacturing contexts, you often have multiple constraints (like materials or time) that must be satisfied simultaneously. The key is to define variables for the unknown quantities and translate the word problem into mathematical relationships that balance all constraints.

4. 3x + 2y = 16; 2x - 3y = -5; x = ? Answer: 2 Solution: Multiply the first equation by 3: 3(3x + 2y) = 3(16) → 9x + 6y = 48 Multiply the second equation by 2: 2(2x - 3y) = 2(-5) → 4x - 6y = -10 Add the two equations: (9x + 6y) + (4x - 6y) = 48 + (-10) → 13x = 38 Solve for x: x = 38/13 → x = 2 The answer is x = 2.
   Full step-by-step solution

   Step 1: Multiply the first equation by 3: 3(3x + 2y) = 3(16) → 9x + 6y = 48 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) = 48 + (-10) → 13x = 38 Step 4: Solve for x: x = 38/13 → x = 2 The answer is x = 2.

5. A technology company is analyzing the optimal production mix for two new smartphone models. The Standard model requires 2 hours of assembly and 1 hour of quality testing, while the Premium model requires 3 hours of assembly and 2 hours of quality testing. The factory has 220 assembly hours and 140 quality testing hours available daily. If the company wants to use all available hours exactly, how many of each model should they produce? Answer: (80, 20) Solution: Let x = number of Standard models, y = number of Premium models Write the assembly hours equation: 2x + 3y = 220 Write the quality testing hours equation: x + 2y = 140 Solve the system using substitution or elimination Multiply the second equation by 2: 2x + 4y = 280 Subtract the first equation…
   Full step-by-step solution

   Step 1: Let x = number of Standard models, y = number of Premium models Step 2: Write the assembly hours equation: 2x + 3y = 220 Step 3: Write the quality testing hours equation: x + 2y = 140 Step 4: Solve the system using substitution or elimination Step 5: Multiply the second equation by 2: 2x + 4y = 280 Step 6: Subtract the first equation from this result: (2x + 4y) - (2x + 3y) = 280 - 220 Step 7: Simplify: y = 60 Step 8: Substitute y = 60 into x + 2y = 140: x + 2(60) = 140 Step 9: Simplify: x + 120 = 140 Step 10: Solve: x = 20 Step 11: Check: 2(20) + 3(60) = 40 + 180 = 220 (assembly) Step 12: Check: 20 + 2(60) = 20 + 120 = 140 (testing) The company should produce 20 Standard models and 60 Premium models.

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (0,8). A circle is inscribed in this triangle such that it touches all three sides. What is the radius of this inscribed circle? Answer: 2 Solution: Calculate the area of the triangle. The base is 6 units and height is 8 units, so area = (1/2) × base × height = (1/2) × 6 × 8 = 24 square units.
   Full step-by-step solution

   Step 1: Calculate the area of the triangle. The base is 6 units and height is 8 units, so area = (1/2) × base × height = (1/2) × 6 × 8 = 24 square units. Step 2: Calculate the hypotenuse using the Pythagorean theorem: sqrt(6² + 8²) = sqrt(36 + 64) = sqrt(100) = 10 units. Step 3: Calculate the perimeter: 6 + 8 + 10 = 24 units. Step 4: For any triangle, the radius r of the inscribed circle is given by r = (2 × area) / perimeter. Step 5: Substitute the values: r = (2 × 24) / 24 = 48 / 24 = 2. The radius of the inscribed circle is 2 units.

7. 2x + 3y = 12; 3x - 2y = 5; x = ? Answer: 3 Solution: 1) 2x + 3y = 12 2) 3x - 2y = 5 We need to solve for x.
   Full step-by-step solution

   We are given the system of equations: 1) 2x + 3y = 12 2) 3x - 2y = 5 We need to solve for x. --- **Step 1: Choose a method** We can use elimination or substitution. Let's use elimination to eliminate y. --- **Step 2: Make coefficients of y equal in magnitude** Multiply equation (1) by 2: 2 * (2x + 3y) = 2 * 12 4x + 6y = 24 ...(3) Multiply equation (2) by 3: 3 * (3x - 2y) = 3 * 5 9x - 6y = 15 ...(4) --- **Step 3: Add equations (3) and (4) to eliminate y** (4x + 6y) + (9x - 6y) = 24 + 15 4x + 9x + 6y - 6y = 39 13x = 39 --- **Step 4: Solve for x** x = 39 / 13 x = 3 --- **Step 5: Conclusion** The value of x is 3. We can check by substituting x = 3 into equation (1): 2(3) + 3y = 12 → 6 + 3y = 12 → 3y = 6 → y = 2. Then check equation (2): 3(3) - 2(2) = 9 - 4 = 5, correct. **Final answer:** x = 3