Sure, let's solve the equation step by step.
We start with the given equation:
[tex]\[ x - 2x - \left(\frac{3x - 4}{7}\right) = \left(\frac{4x - 27}{3}\right) - 3 \][/tex]
1. Combine like terms on the left-hand side:
[tex]\[ x - 2x = -x \][/tex]
So the equation becomes:
[tex]\[ -x - \left(\frac{3x - 4}{7}\right) = \left(\frac{4x - 27}{3}\right) - 3 \][/tex]
2. Clear the fractions by finding a common denominator for the entire equation.
- The common denominator for the terms involving fractions can be found from 7 and 3, which is 21.
3. Rewrite the equation by multiplying all terms by 21 to eliminate the fractions:
[tex]\[ 21 \left(-x - \frac{3x - 4}{7}\right) = 21 \left(\frac{4x - 27}{3} - 3\right) \][/tex]
4. Distribute 21 inside the parentheses:
[tex]\[ 21(-x) - 21 \left(\frac{3x - 4}{7}\right) = 21 \left(\frac{4x - 27}{3}\right) - 21 \cdot 3 \][/tex]
5. Simplify each term:
[tex]\[ 21(-x) - 3(3x - 4) = 7(4x - 27) - 63 \][/tex]
6. Distribute inside the parentheses:
[tex]\[ -21x - (9x - 12) = 28x - 189 - 63 \][/tex]
7. Simplify both sides of the equation:
[tex]\[ -21x - 9x + 12 = 28x - 252 \][/tex]
8. Combine like terms:
[tex]\[ -30x + 12 = 28x - 252 \][/tex]
9. Move all terms involving [tex]\( x \)[/tex] to one side and constant terms to the other side:
[tex]\[ -30x - 28x = -252 - 12 \][/tex]
[tex]\[ -58x = -264 \][/tex]
10. Solve for [tex]\( x \)[/tex] by dividing both sides by -58:
[tex]\[ x = \frac{-264}{-58} \][/tex]
[tex]\[ x = \frac{264}{58} \][/tex]
[tex]\[ x = \frac{132}{29} \][/tex]
So the solution to the given equation is:
[tex]\[ x = \frac{132}{29} \][/tex]
