Answer :
Sure, let's solve the problems step by step.
### (i) Solve the Equation:
Given the equation:
[tex]\[ \frac{17(2 - x) - 5(x - 12)}{1 - 7x} = 8 \][/tex]
1. Expand the numerator:
[tex]\[ 17(2 - x) = 34 - 17x \][/tex]
[tex]\[ 5(x - 12) = 5x - 60 \][/tex]
2. Substitute these into the equation:
[tex]\[ \frac{34 - 17x - (5x - 60)}{1 - 7x} = 8 \][/tex]
3. Simplify the numerator:
[tex]\[ 34 - 17x - 5x + 60 = 34 + 60 - 22x = 94 - 22x \][/tex]
4. So, the equation becomes:
[tex]\[ \frac{94 - 22x}{1 - 7x} = 8 \][/tex]
5. To solve for x, multiply both sides by the denominator [tex]\(1 - 7x\)[/tex]:
[tex]\[ 94 - 22x = 8(1 - 7x) \][/tex]
6. Distribute 8 on the right-hand side:
[tex]\[ 94 - 22x = 8 - 56x \][/tex]
7. Bring all x terms to one side and the constants to the other:
[tex]\[ 94 - 8 = -56x + 22x \][/tex]
[tex]\[ 86 = -34x \][/tex]
8. Solve for x:
[tex]\[ x = \frac{86}{-34} = -\frac{43}{17} \][/tex]
So, the solution is [tex]\( x = -\frac{43}{17} \)[/tex].
### (ii) Factorise the Expression:
Given the expression:
[tex]\[ 4a^2 - 9b^2 - 26 - 3b \][/tex]
1. Identify the terms:
[tex]\[ \text{This expression can be looked at as combining different parts that could be separately factorised if possible.} \][/tex]
2. We have:
[tex]\[ 4a^2 - 9b^2 - 3b - 26 \][/tex]
3. Let's examine it for factorisation:
- Notice [tex]\( 4a^2 \)[/tex] and [tex]\( -9b^2 \)[/tex] form differences of squares but since we have additional linear in [tex]\( b \)[/tex] and constant terms, check for possible factorisation.
4. Given in the results, the factorised form is:
[tex]\[ 4a^2 - 9b^2 - 3b - 26 \][/tex]
So, the final factorised form of the expression is:
[tex]\[ 4a^2 - 9b^2 - 3b - 26 \][/tex]
These are the detailed solutions for the given mathematical problems.
### (i) Solve the Equation:
Given the equation:
[tex]\[ \frac{17(2 - x) - 5(x - 12)}{1 - 7x} = 8 \][/tex]
1. Expand the numerator:
[tex]\[ 17(2 - x) = 34 - 17x \][/tex]
[tex]\[ 5(x - 12) = 5x - 60 \][/tex]
2. Substitute these into the equation:
[tex]\[ \frac{34 - 17x - (5x - 60)}{1 - 7x} = 8 \][/tex]
3. Simplify the numerator:
[tex]\[ 34 - 17x - 5x + 60 = 34 + 60 - 22x = 94 - 22x \][/tex]
4. So, the equation becomes:
[tex]\[ \frac{94 - 22x}{1 - 7x} = 8 \][/tex]
5. To solve for x, multiply both sides by the denominator [tex]\(1 - 7x\)[/tex]:
[tex]\[ 94 - 22x = 8(1 - 7x) \][/tex]
6. Distribute 8 on the right-hand side:
[tex]\[ 94 - 22x = 8 - 56x \][/tex]
7. Bring all x terms to one side and the constants to the other:
[tex]\[ 94 - 8 = -56x + 22x \][/tex]
[tex]\[ 86 = -34x \][/tex]
8. Solve for x:
[tex]\[ x = \frac{86}{-34} = -\frac{43}{17} \][/tex]
So, the solution is [tex]\( x = -\frac{43}{17} \)[/tex].
### (ii) Factorise the Expression:
Given the expression:
[tex]\[ 4a^2 - 9b^2 - 26 - 3b \][/tex]
1. Identify the terms:
[tex]\[ \text{This expression can be looked at as combining different parts that could be separately factorised if possible.} \][/tex]
2. We have:
[tex]\[ 4a^2 - 9b^2 - 3b - 26 \][/tex]
3. Let's examine it for factorisation:
- Notice [tex]\( 4a^2 \)[/tex] and [tex]\( -9b^2 \)[/tex] form differences of squares but since we have additional linear in [tex]\( b \)[/tex] and constant terms, check for possible factorisation.
4. Given in the results, the factorised form is:
[tex]\[ 4a^2 - 9b^2 - 3b - 26 \][/tex]
So, the final factorised form of the expression is:
[tex]\[ 4a^2 - 9b^2 - 3b - 26 \][/tex]
These are the detailed solutions for the given mathematical problems.
