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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6. Given that y is directly proportional to x and that y=30 when x=4 a) Find an equation in terms of x and y. Step 1: Express the direct proportionality. y = kx where k is the constant of proportionality. Step 2: Use the given values y=30 and x=4 to find k. 30 = k(4) k = (30)/(4) k = (15)/(2) Step 3: Write the equation in terms of x and y. y = (15)/(2)x The final answer is y = (15)/(2)x. b) Determine the value of y when x = (2)/(3). Step 1: Use the equation found in part a). y = (15)/(2)x Step 2: Substitute x = (2)/(3) into the equation. y = (15)/(2) ((2)/(3)) y = (15 × 2)/(2 × 3) y = (30)/(6) y = 5 The final answer is 5. 7. Solve a) -2(x-3) 7-5x, and represent your solution on the real number line. Step 1: Distribute the -2 on the left side. -2x + 6 7 - 5x Step 2: Add 5x to both sides. -2x + 5x + 6 7 3x + 6 7 Step 3: Subtract 6 from both sides. 3x 7 - 6 3x 1 Step 4: Divide by 3. x (1)/(3) The solution is x (1)/(3). Step 5: Represent the solution on the real number line. To represent x (1)/(3) on a number line, draw a closed circle at (1)/(3) and shade the line to the left of (1)/(3). ` <------------------•------------------> 1/3 ` b) |2x-5| = 3 Step 1: Set up two separate equations based on the definition of absolute value. Case 1: 2x - 5 = 3 Case 2: 2x - 5 = -3 Step 2: Solve Case 1. 2x - 5 = 3 2x = 3 + 5 2x = 8 x = (8)/(2) x = 4 Step 3: Solve Case 2. 2x - 5 = -3 2x = -3 + 5 2x = 2 x = (2)/(2) x = 1 The solutions are x=4 or x=1. 8. Given that OP = (a+5)i + 2j and OQ = 2ai - 2j, where a is a constant. a) Find PQ in terms of a. Step 1: Use the vector subtraction formula PQ = OQ - OP. PQ = (2ai - 2j) - ((a+5)i + 2j) Step 2: Group the i components and the j components. PQ = (2a - (a+5))i + (-2 - 2)j Step 3: Simplify the components. PQ = (2a - a - 5)i + (-4)j PQ = (a - 5)i - 4j The final answer is (a - 5)i - 4j. b) When PQ = 3i - 4j, find a. Step 1: Equate the expression for PQ from part a) with the given vector. (a - 5)i - 4j = 3i - 4j Step 2: Equate the corresponding i components. a - 5 = 3 Step 3: Solve for a. a = 3 + 5 a = 8 (The j components, -4 = -4, are consistent.) The final answer is 8.