Answer :
Let's solve each equation step-by-step.
### (a) [tex]\( x - 1 = \frac{1}{2} x \)[/tex]
1. Subtract [tex]\(\frac{1}{2} x\)[/tex] from both sides:
[tex]\[ x - \frac{1}{2} x - 1 = 0 \][/tex]
2. Simplify the left side:
[tex]\[ \frac{1}{2} x - 1 = 0 \][/tex]
3. Add 1 to both sides:
[tex]\[ \frac{1}{2} x = 1 \][/tex]
4. Multiply both sides by 2:
[tex]\[ x = 2 \][/tex]
So, the solution for (a) is [tex]\( x = 2 \)[/tex].
### (b) [tex]\( 2(x - 1) + 3(x + 1) = 4(x + 4) \)[/tex]
1. Expand both sides:
[tex]\[ 2x - 2 + 3x + 3 = 4x + 16 \][/tex]
2. Combine like terms:
[tex]\[ 5x + 1 = 4x + 16 \][/tex]
3. Subtract 4x from both sides:
[tex]\[ x + 1 = 16 \][/tex]
4. Subtract 1 from both sides:
[tex]\[ x = 15 \][/tex]
So, the solution for (b) is [tex]\( x = 15 \)[/tex].
### (c) [tex]\( 2 y - [7 - (5 y - 4)] = 6 \)[/tex]
1. Distribute and simplify inside the brackets:
[tex]\[ 2y - [7 - 5y + 4] = 6 \][/tex]
2. Combine like terms inside the brackets:
[tex]\[ 2y - (11 - 5y) = 6 \][/tex]
3. Distribute the negative sign:
[tex]\[ 2y - 11 + 5y = 6 \][/tex]
4. Combine like terms:
[tex]\[ 7y - 11 = 6 \][/tex]
5. Add 11 to both sides:
[tex]\[ 7y = 17 \][/tex]
6. Divide both sides by 7:
[tex]\[ y = \frac{17}{7} \][/tex]
So, the solution for (c) is [tex]\( y = \frac{17}{7} \)[/tex].
### (d) [tex]\( \frac{3}{4} x - 5 = 0.5 x \)[/tex]
1. Rewrite [tex]\( 0.5 x \)[/tex] as [tex]\( \frac{1}{2} x \)[/tex]:
[tex]\[ \frac{3}{4} x - 5 = \frac{1}{2} x \][/tex]
2. Subtract [tex]\( \frac{1}{2} x \)[/tex] from both sides:
[tex]\[ \frac{3}{4} x - \frac{1}{2} x - 5 = 0 \][/tex]
3. Find a common denominator to subtract [tex]\( \frac{1}{2} x \)[/tex]:
[tex]\[ \frac{3}{4} x - \frac{2}{4} x - 5 = 0 \][/tex]
4. Combine like terms:
[tex]\[ \frac{1}{4} x - 5 = 0 \][/tex]
5. Add 5 to both sides:
[tex]\[ \frac{1}{4} x = 5 \][/tex]
6. Multiply both sides by 4:
[tex]\[ x = 20 \][/tex]
So, the solution for (d) is [tex]\( x = 20 \)[/tex].
### (e) [tex]\( \frac{2 y + 7}{4} = 12 \)[/tex]
1. Multiply both sides by 4:
[tex]\[ 2y + 7 = 48 \][/tex]
2. Subtract 7 from both sides:
[tex]\[ 2y = 41 \][/tex]
3. Divide both sides by 2:
[tex]\[ y = \frac{41}{2} \][/tex]
So, the solution for (e) is [tex]\( y = \frac{41}{2} \)[/tex].
### (f) [tex]\( \frac{4 y - 1}{5 y + 1} = \frac{5}{7} \)[/tex]
1. Cross-multiply to get rid of the fractions:
[tex]\[ 7(4y - 1) = 5(5y + 1) \][/tex]
2. Distribute both sides:
[tex]\[ 28y - 7 = 25y + 5 \][/tex]
3. Subtract [tex]\( 25y \)[/tex] from both sides:
[tex]\[ 3y - 7 = 5 \][/tex]
4. Add 7 to both sides:
[tex]\[ 3y = 12 \][/tex]
5. Divide both sides by 3:
[tex]\[ y = 4 \][/tex]
So, the solution for (f) is [tex]\( y = 4 \)[/tex].
### (g) [tex]\( \frac{a + 1}{4} + \frac{a - 1}{3} = 4 \)[/tex]
1. Find a common denominator and combine the fractions:
[tex]\[ \frac{3(a + 1) + 4(a - 1)}{12} = 4 \][/tex]
2. Distribute and simplify inside the numerator:
[tex]\[ \frac{3a + 3 + 4a - 4}{12} = 4 \][/tex]
3. Combine like terms:
[tex]\[ \frac{7a - 1}{12} = 4 \][/tex]
4. Multiply both sides by 12:
[tex]\[ 7a - 1 = 48 \][/tex]
5. Add 1 to both sides:
[tex]\[ 7a = 49 \][/tex]
6. Divide both sides by 7:
[tex]\[ a = 7 \][/tex]
So, the solution for (g) is [tex]\( a = 7 \)[/tex].
### (h) [tex]\( \frac{b - 4}{3} - \frac{2 b + 1}{6} = \frac{5 b - 1}{2} \)[/tex]
1. Find a common denominator for the fractions on the left side (common denominator is 6):
[tex]\[ \frac{2(b - 4) - (2b + 1)}{6} = \frac{5b - 1}{2} \][/tex]
2. Distribute the 2 and simplify:
[tex]\[ \frac{2b - 8 - 2b - 1}{6} = \frac{5b - 1}{2} \][/tex]
3. Combine like terms:
[tex]\[ \frac{-9}{6} = \frac{5b - 1}{2} \][/tex]
4. Simplify [tex]\(\frac{-9}{6}\)[/tex] to [tex]\(-\frac{3}{2}\)[/tex]:
[tex]\[ -\frac{3}{2} = \frac{5b - 1}{2} \][/tex]
5. Get rid of the fractions by multiplying both sides by 2:
[tex]\[ -3 = 5b - 1 \][/tex]
6. Add 1 to both sides:
[tex]\[ -2 = 5b \][/tex]
7. Divide both sides by 5:
[tex]\[ b = -\frac{2}{5} \][/tex]
So, the solution for (h) is [tex]\( b = -\frac{2}{5} \)[/tex].
In summary:
(a) [tex]\( x = 2 \)[/tex]
(b) [tex]\( x = 15 \)[/tex]
(c) [tex]\( y = \frac{17}{7} \)[/tex]
(d) [tex]\( x = 20 \)[/tex]
(e) [tex]\( y = \frac{41}{2} \)[/tex]
(f) [tex]\( y = 4 \)[/tex]
(g) [tex]\( a = 7 \)[/tex]
(h) [tex]\( b = -\frac{2}{5} \)[/tex]
### (a) [tex]\( x - 1 = \frac{1}{2} x \)[/tex]
1. Subtract [tex]\(\frac{1}{2} x\)[/tex] from both sides:
[tex]\[ x - \frac{1}{2} x - 1 = 0 \][/tex]
2. Simplify the left side:
[tex]\[ \frac{1}{2} x - 1 = 0 \][/tex]
3. Add 1 to both sides:
[tex]\[ \frac{1}{2} x = 1 \][/tex]
4. Multiply both sides by 2:
[tex]\[ x = 2 \][/tex]
So, the solution for (a) is [tex]\( x = 2 \)[/tex].
### (b) [tex]\( 2(x - 1) + 3(x + 1) = 4(x + 4) \)[/tex]
1. Expand both sides:
[tex]\[ 2x - 2 + 3x + 3 = 4x + 16 \][/tex]
2. Combine like terms:
[tex]\[ 5x + 1 = 4x + 16 \][/tex]
3. Subtract 4x from both sides:
[tex]\[ x + 1 = 16 \][/tex]
4. Subtract 1 from both sides:
[tex]\[ x = 15 \][/tex]
So, the solution for (b) is [tex]\( x = 15 \)[/tex].
### (c) [tex]\( 2 y - [7 - (5 y - 4)] = 6 \)[/tex]
1. Distribute and simplify inside the brackets:
[tex]\[ 2y - [7 - 5y + 4] = 6 \][/tex]
2. Combine like terms inside the brackets:
[tex]\[ 2y - (11 - 5y) = 6 \][/tex]
3. Distribute the negative sign:
[tex]\[ 2y - 11 + 5y = 6 \][/tex]
4. Combine like terms:
[tex]\[ 7y - 11 = 6 \][/tex]
5. Add 11 to both sides:
[tex]\[ 7y = 17 \][/tex]
6. Divide both sides by 7:
[tex]\[ y = \frac{17}{7} \][/tex]
So, the solution for (c) is [tex]\( y = \frac{17}{7} \)[/tex].
### (d) [tex]\( \frac{3}{4} x - 5 = 0.5 x \)[/tex]
1. Rewrite [tex]\( 0.5 x \)[/tex] as [tex]\( \frac{1}{2} x \)[/tex]:
[tex]\[ \frac{3}{4} x - 5 = \frac{1}{2} x \][/tex]
2. Subtract [tex]\( \frac{1}{2} x \)[/tex] from both sides:
[tex]\[ \frac{3}{4} x - \frac{1}{2} x - 5 = 0 \][/tex]
3. Find a common denominator to subtract [tex]\( \frac{1}{2} x \)[/tex]:
[tex]\[ \frac{3}{4} x - \frac{2}{4} x - 5 = 0 \][/tex]
4. Combine like terms:
[tex]\[ \frac{1}{4} x - 5 = 0 \][/tex]
5. Add 5 to both sides:
[tex]\[ \frac{1}{4} x = 5 \][/tex]
6. Multiply both sides by 4:
[tex]\[ x = 20 \][/tex]
So, the solution for (d) is [tex]\( x = 20 \)[/tex].
### (e) [tex]\( \frac{2 y + 7}{4} = 12 \)[/tex]
1. Multiply both sides by 4:
[tex]\[ 2y + 7 = 48 \][/tex]
2. Subtract 7 from both sides:
[tex]\[ 2y = 41 \][/tex]
3. Divide both sides by 2:
[tex]\[ y = \frac{41}{2} \][/tex]
So, the solution for (e) is [tex]\( y = \frac{41}{2} \)[/tex].
### (f) [tex]\( \frac{4 y - 1}{5 y + 1} = \frac{5}{7} \)[/tex]
1. Cross-multiply to get rid of the fractions:
[tex]\[ 7(4y - 1) = 5(5y + 1) \][/tex]
2. Distribute both sides:
[tex]\[ 28y - 7 = 25y + 5 \][/tex]
3. Subtract [tex]\( 25y \)[/tex] from both sides:
[tex]\[ 3y - 7 = 5 \][/tex]
4. Add 7 to both sides:
[tex]\[ 3y = 12 \][/tex]
5. Divide both sides by 3:
[tex]\[ y = 4 \][/tex]
So, the solution for (f) is [tex]\( y = 4 \)[/tex].
### (g) [tex]\( \frac{a + 1}{4} + \frac{a - 1}{3} = 4 \)[/tex]
1. Find a common denominator and combine the fractions:
[tex]\[ \frac{3(a + 1) + 4(a - 1)}{12} = 4 \][/tex]
2. Distribute and simplify inside the numerator:
[tex]\[ \frac{3a + 3 + 4a - 4}{12} = 4 \][/tex]
3. Combine like terms:
[tex]\[ \frac{7a - 1}{12} = 4 \][/tex]
4. Multiply both sides by 12:
[tex]\[ 7a - 1 = 48 \][/tex]
5. Add 1 to both sides:
[tex]\[ 7a = 49 \][/tex]
6. Divide both sides by 7:
[tex]\[ a = 7 \][/tex]
So, the solution for (g) is [tex]\( a = 7 \)[/tex].
### (h) [tex]\( \frac{b - 4}{3} - \frac{2 b + 1}{6} = \frac{5 b - 1}{2} \)[/tex]
1. Find a common denominator for the fractions on the left side (common denominator is 6):
[tex]\[ \frac{2(b - 4) - (2b + 1)}{6} = \frac{5b - 1}{2} \][/tex]
2. Distribute the 2 and simplify:
[tex]\[ \frac{2b - 8 - 2b - 1}{6} = \frac{5b - 1}{2} \][/tex]
3. Combine like terms:
[tex]\[ \frac{-9}{6} = \frac{5b - 1}{2} \][/tex]
4. Simplify [tex]\(\frac{-9}{6}\)[/tex] to [tex]\(-\frac{3}{2}\)[/tex]:
[tex]\[ -\frac{3}{2} = \frac{5b - 1}{2} \][/tex]
5. Get rid of the fractions by multiplying both sides by 2:
[tex]\[ -3 = 5b - 1 \][/tex]
6. Add 1 to both sides:
[tex]\[ -2 = 5b \][/tex]
7. Divide both sides by 5:
[tex]\[ b = -\frac{2}{5} \][/tex]
So, the solution for (h) is [tex]\( b = -\frac{2}{5} \)[/tex].
In summary:
(a) [tex]\( x = 2 \)[/tex]
(b) [tex]\( x = 15 \)[/tex]
(c) [tex]\( y = \frac{17}{7} \)[/tex]
(d) [tex]\( x = 20 \)[/tex]
(e) [tex]\( y = \frac{41}{2} \)[/tex]
(f) [tex]\( y = 4 \)[/tex]
(g) [tex]\( a = 7 \)[/tex]
(h) [tex]\( b = -\frac{2}{5} \)[/tex]
