Simplify: a) (y2+4y+3)/(2y2+7y+5) b) (2k−2t)/15 ÷ (k-t)/3 c) (x+4)/2 − (2x+1)/3

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Here are the solutions to the questions. QUESTION FOUR a) Simplify: (y²+4y+3)/(2y²+7y+5) Step 1: Factorize the numerator. To factor y²+4y+3, find two numbers that multiply to 3 and add to 4. These numbers are 1 and 3. y²+4y+3 = (y+1)(y+3) Step 2: Factorize the denominator. To factor 2y²+7y+5, find two numbers that multiply to 2 × 5 = 10 and add to 7. These numbers are 2 and 5. 2y²+7y+5 = 2y²+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 x²+8x+7, as x+8x+7 simplifies to 9x+7 which is a linear expression and cannot be factorised into binomials. Step 1: For x²+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. x²+8x+7 = (x+1)(x+7) The factorised expression is (x+1)(x+7). ii. 6 − 5b − 4b² Step 1: Rearrange the expression in standard quadratic form (ab²+bb+c). −4b² − 5b + 6 Step 2: Factor out −1 to make the leading coefficient positive. −(4b² + 5b − 6) Step 3: Factor the quadratic expression 4b² + 5b − 6. Find two numbers that multiply to 4 × (−6) = −24 and add to 5. These numbers are 8 and −3. 4b² + 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).

Simplify: a) (y2+4y+3)/(2y2+7y+5) b) (2k-2t)/15 ÷ (k-t)/3 c) (x+4)/2 - (2x+1)/3

Answers

Step-by-step solution

Part a· 3 steps

Factorize the numerator.

Factorize the denominator.

Simplify the fraction by canceling common factors.

Part b· 4 steps
$\frac{2}{5}$

Rewrite the division as multiplication by the reciprocal of the second fraction.

Factorize the numerator of the first fraction.

Multiply the fractions and simplify by canceling common factors.

Reduce the fraction to its simplest form.

Part c· 4 steps
$x=10$

Find a common denominator for the fractions, which is 6.

Multiply each term by the common denominator to eliminate the fractions.

Expand and simplify the equation.

Solve for $x$ .

Part d· 4 steps
$x=1, y=-2$

Label the equations.

From equation (2), express $x$ in terms of $y$ .

Substitute this expression for $x$ into equation (1).

Substitute the value of $y$ back into the expression for $x$ from Step 2.

Part e(i)· 2 steps(x+1)(x+7)

For $x^2+8x+7$ , find two numbers that multiply to 7 and add to 8.

Write the factored form.

Part e(ii)· 4 steps-(4b-3)(b+2)

Rearrange the expression in standard quadratic form ( $ab^2+bb+c$ ).

Factor out $-1$ to make the leading coefficient positive.

Factor the quadratic expression $4b^2 + 5b - 6$ .

Combine with the factored out $-1$ .

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