Certainly! Let's solve these equations using the transposition method step by step.
### Equation (a)
[tex]\[ 3x - \frac{1}{4} = \frac{1}{6} - x \][/tex]
1. Combine like terms involving [tex]\( x \)[/tex] on one side:
Add [tex]\( x \)[/tex] to both sides:
[tex]\[ 3x + x - \frac{1}{4} = \frac{1}{6} - x + x \][/tex]
Simplify to:
[tex]\[ 4x - \frac{1}{4} = \frac{1}{6} \][/tex]
2. Move the constant term (-1/4) to the other side:
Add [tex]\( \frac{1}{4} \)[/tex] to both sides:
[tex]\[ 4x - \frac{1}{4} + \frac{1}{4} = \frac{1}{6} + \frac{1}{4} \][/tex]
Simplify to:
[tex]\[ 4x = \frac{1}{6} + \frac{1}{4} \][/tex]
3. Combine the fractions on the right side:
Find a common denominator for [tex]\( \frac{1}{6} \)[/tex] and [tex]\( \frac{1}{4} \)[/tex], which is 12:
[tex]\[ \frac{1}{6} = \frac{2}{12} \][/tex]
[tex]\[ \frac{1}{4} = \frac{3}{12} \][/tex]
Add them together:
[tex]\[ \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \][/tex]
Now the equation is:
[tex]\[ 4x = \frac{5}{12} \][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides by 4 (or multiply by [tex]\( \frac{1}{4} \)[/tex]):
[tex]\[ x = \frac{5}{12} \times \frac{1}{4} = \frac{5}{48} \][/tex]
Therefore, the solution to equation (a) is:
[tex]\[ x = \frac{5}{48} \][/tex]
### Equation (c)
[tex]\[ 4x + 3(x + 3) = 29 + 3x \][/tex]
1. Distribute the 3 in the left-hand side expression:
[tex]\[ 4x + 3(x + 3) = 4x + 3x + 9 \][/tex]
So the equation becomes:
[tex]\[ 4x + 3x + 9 = 29 + 3x \][/tex]
Simplify the left-hand side:
[tex]\[ 7x + 9 = 29 + 3x \][/tex]
2. Combine like terms involving [tex]\( x \)[/tex] on one side:
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ 7x - 3x + 9 = 29 + 3x - 3x \][/tex]
Simplify to:
[tex]\[ 4x + 9 = 29 \][/tex]
3. Move the constant term (9) to the other side:
Subtract 9 from both sides:
[tex]\[ 4x + 9 - 9 = 29 - 9 \][/tex]
Simplify to:
[tex]\[ 4x = 20 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides by 4:
[tex]\[ x = \frac{20}{4} = 5 \][/tex]
Therefore, the solution to equation (c) is:
[tex]\[ x = 5 \][/tex]
In summary, the solutions to the equations are:
(a) [tex]\( x = \frac{5}{48} \)[/tex]
(c) [tex]\( x = 5 \)[/tex]
