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 questions. QUESTION FOUR a) Simplify: (y^2+4y+3)/(2y^2+7y+5) Step 1: Factorize the numerator. To factor y^2+4y+3, find two numbers that multiply to 3 and add to 4. These numbers are 1 and 3. y^2+4y+3 = (y+1)(y+3) Step 2: Factorize the denominator. To factor 2y^2+7y+5, find two numbers that multiply to 2 × 5 = 10 and add to 7. These numbers 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: Simplify the fraction 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: Rewrite the division as multiplication by the reciprocal of the second fraction. (2k-2t)/(15) × (3)/(k-t) Step 2: Factorize the numerator of the first fraction. (2(k-t))/(15) × (3)/(k-t) Step 3: Multiply the fractions and simplify by canceling common factors. (2(k-t) × 3)/(15 × (k-t)) = (6(k-t))/(15(k-t)) Assuming k ≠ t, we can cancel (k-t). = (6)/(15) Step 4: Reduce the fraction to its simplest form. (6)/(15) = (2 × 3)/(5 × 3) = (2)/(5) The expression as a single fraction is (2)/(5). c) Solve: (x+4)/(2) - (2x+1)/(3) = 0 Step 1: Find a common denominator for the fractions, which is 6. Step 2: Multiply each term by the common denominator to eliminate the fractions. 6 ((x+4)/(2)) - 6 ((2x+1)/(3)) = 6(0) 3(x+4) - 2(2x+1) = 0 Step 3: Expand and simplify the equation. 3x+12 - 4x-2 = 0 -x+10 = 0 Step 4: Solve for x. -x = -10 x = 10 The solution is x=10. d) Solve the simultaneous equations: 2x+3y=-4 and x-y=3 Step 1: Label the equations. (1) 2x+3y=-4 (2) x-y=3 Step 2: From equation (2), express x in terms of y. x = y+3 Step 3: Substitute this expression for x into equation (1). 2(y+3)+3y=-4 2y+6+3y=-4 5y+6=-4 5y = -4-6 5y = -10 y = (-10)/(5) y = -2 Step 4: Substitute the value of y back into the expression for x from Step 2. x = -2+3 x = 1 The solution is x=1, y=-2. e) Factorise the following expressions; i. x+8x+7 Assumption: The expression is intended to be x^2+8x+7, as x+8x+7 simplifies to 9x+7 which is a linear expression and cannot be factorised into binomials. Step 1: For x^2+8x+7, find two numbers that multiply to 7 and add to 8. These numbers are 1 and 7. Step 2: Write the factored form. x^2+8x+7 = (x+1)(x+7) The factorised expression is (x+1)(x+7). ii. 6 - 5b - 4b^2 Step 1: Rearrange the expression in standard quadratic form (ab^2+bb+c). -4b^2 - 5b + 6 Step 2: Factor out -1 to make the leading coefficient positive. -(4b^2 + 5b - 6) Step 3: Factor the quadratic expression 4b^2 + 5b - 6. Find two numbers that multiply to 4 × (-6) = -24 and add to 5. These numbers are 8 and -3. 4b^2 + 8b - 3b - 6 4b(b+2) - 3(b+2) (4b-3)(b+2) Step 4: Combine with the factored out -1. -(4b-3)(b+2) This can also be written as (3-4b)(b+2). The factorised expression is -(4b-3)(b+2) or (3-4b)(b+2). 3 done, 2 left today. You're making progress.