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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Question Four a) Simplify: (y^2+4y+3)/(2y^2+7y+5) Step 1: Factorize the numerator. We need two numbers that multiply to 3 and add to 4. These are 1 and 3. y^2+4y+3 = (y+1)(y+3) Step 2: Factorize the denominator. We need two numbers that multiply to 2 × 5 = 10 and add to 7. These are 2 and 5. 2y^2+7y+5 = 2y^2+2y+5y+5 = 2y(y+1)+5(y+1) = (2y+5)(y+1) Step 3: Substitute the factored expressions back into the fraction and simplify by canceling common factors. ((y+1)(y+3))/((2y+5)(y+1)) = (y+3)/(2y+5) The simplified expression is (y+3)/(2y+5). b) Express as a single fraction: (2k-2t)/(15) + (k-t)/(3) Step 1: Find a common denominator, which is 15. Multiply the second fraction by (5)/(5). (2k-2t)/(15) + (5(k-t))/(5 × 3) = (2k-2t)/(15) + (5k-5t)/(15) Step 2: Combine the numerators over the common denominator. ((2k-2t) + (5k-5t))/(15) = (2k-2t+5k-5t)/(15) Step 3: Combine like terms in the numerator. (7k-7t)/(15) The expression as a single fraction is (7k-7t)/(15). c) Solve: (x+4)/(2) - (2x+1)/(3) = 0 Step 1: Find the least common multiple (LCM) of the denominators 2 and 3, which is 6. Multiply the entire equation by 6. 6 ((x+4)/(2)) - 6 ((2x+1)/(3)) = 6 × 0 3(x+4) - 2(2x+1) = 0 Step 2: Distribute the numbers into the parentheses. 3x+12 - (4x+2) = 0 3x+12-4x-2 = 0 Step 3: Combine like terms. -x+10 = 0 Step 4: Solve for x. 10 = x The solution is x=10. d) Solve the simultaneous equations: 2x + 3y = -4 (Equation 1) x - y = 3 (Equation 2) Step 1: From Equation 2, express x in terms of y. x = y+3 (Equation 3) Step 2: Substitute Equation 3 into Equation 1. 2(y+3) + 3y = -4 2y+6+3y = -4 Step 3: Combine like terms and solve for y. 5y+6 = -4 5y = -4-6 5y = -10 y = (-10)/(5) y = -2 Step 4: Substitute the value of y back into Equation 3 to find x. x = -2+3 x = 1 The solution is x=1, y=-2. e) Factorise the following expressions: i. x^2 + 8x + 7 Step 1: Find two numbers that multiply to 7 and add to 8. These numbers are 1 and 7. x^2 + 8x + 7 = (x+1)(x+7) The factored expression is (x+1)(x+7). ii. 6 - 5b - 4b^2 Step 1: Rearrange the expression in standard quadratic form (descending powers of b). -4b^2 - 5b + 6 Step 2: Factor out -1 to make the leading coefficient positive. -(4b^2 + 5b - 6) Step 3: Factorize the quadratic expression 4b^2 + 5b - 6. We look for two numbers that multiply to 4 × (-6) = -24 and add to 5. These numbers are 8 and -3. Rewrite the middle term: 4b^2 + 8b - 3b - 6 Factor by grouping: 4b(b+2) - 3(b+2) (4b-3)(b+2) Step 4: Combine with the factored-out -1. -(4b-3)(b+2) Alternatively, distribute the negative sign into one of the factors, for example, (3-4b)(b+2). The factored expression is (3-4b)(b+2). Got more? Send 'em!