This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems. 1. Step 1: Identify the type of equation. The equation y = 3x + 5 is a linear equation in two variables (x and y). Step 2: Determine the number of solutions for a linear equation in two variables. A linear equation in two variables represents a straight line in a coordinate plane. Every point on this line is a solution to the equation. Since a straight line consists of an infinite number of points, a linear equation in two variables has infinitely many solutions. Step 3: Select the correct option. Therefore, option (iii) is true. The equation y = 3x + 5 has (iii) infinitely many solutions. 2. We need to write four solutions for each equation. 2(i) 2x + y = 7 Step 1: Express y in terms of x. y = 7 - 2x Step 2: Choose four arbitrary values for x and calculate the corresponding y values. • If x = 0, y = 7 - 2(0) = 7. Solution: (0, 7) • If x = 1, y = 7 - 2(1) = 5. Solution: (1, 5) • If x = 2, y = 7 - 2(2) = 3. Solution: (2, 3) • If x = 3, y = 7 - 2(3) = 1. Solution: (3, 1) Four solutions are: (0, 7), (1, 5), (2, 3), (3, 1) 2(ii) x + y = 9 Step 1: Express y in terms of x. y = 9 - x Step 2: Choose four arbitrary values for x and calculate the corresponding y values. • If x = 0, y = 9 - (0) = 9. Solution: (0, 9) • If x = 1, y = 9 - (1) = 9 - . Solution: (1, 9 - ) • If x = 2, y = 9 - (2) = 9 - 2. Solution: (2, 9 - 2) • If x = -1, y = 9 - (-1) = 9 + . Solution: (-1, 9 + ) Four solutions are: (0, 9), (1, 9 - ), (2, 9 - 2), (-1, 9 + ) 2(iii) x = 4y Step 1: Express x in terms of y. (Already done) x = 4y Step 2: Choose four arbitrary values for y and calculate the corresponding x values. • If y = 0, x = 4(0) = 0. Solution: (0, 0) • If y = 1, x = 4(1) = 4. Solution: (4, 1) • If y = 2, x = 4(2) = 8. Solution: (8, 2) • If y = -1, x = 4(-1) = -4. Solution: (-4, -1) Four solutions are: (0, 0), (4, 1), (8, 2), (-4, -1) 3. We need to check which of the given points are solutions to the equation x - 2y = 4. 3(i) (0, 2) Step 1: Substitute x=0 and y=2 into the equation. 0 - 2(2) = 0 - 4 = -4 Step 2: Compare the result with the right side of the equation. Since -4 ≠ 4, (0, 2) is not a solution. 3(ii) (2, 0) Step 1: Substitute x=2 and y=0 into the equation. 2 - 2(0) = 2 - 0 = 2 Step 2: Compare the result with the right side of the equation. Since 2 ≠ 4, (2, 0) is not a solution. 3(iii) (4, 0) Step 1: Substitute x=4 and y=0 into the equation. 4 - 2(0) = 4 - 0 = 4 Step 2: Compare the result with the right side of the equation. Since 4 = 4, (4, 0) is a solution. 3(iv) (sqrt(2), 4sqrt(2)) Step 1: Substitute x=sqrt(2) and y=4sqrt(2) into the equation. sqrt(2) - 2(4sqrt(2)) = sqrt(2) - 8sqrt(2) = -7sqrt(2) Step 2: Compare the result with the right side of the equation. Since -7sqrt(2) ≠ 4, (sqrt(2), 4sqrt(2)) is not a solution. 3(v) (1, 1) Step 1: Substitute x=1 and y=1 into the equation. 1 - 2(1) = 1 - 2 = -1 Step 2: Compare the result with the right side of the equation. Since -1 ≠ 4, (1, 1) is not a solution. Here's a summary of the solutions for x - 2y = 4: • (0, 2) is not a solution. • (2, 0) is not a solution. • (4, 0) is a solution. • (sqrt(2), 4sqrt(2)) is not a solution. • (1, 1) is not a solution. Send me the next one 📸