Answer :
Sure! I'll walk through each equation step-by-step to show how to solve them.
### Equation 7: [tex]\(\frac{x-4}{3} - 5 = 0\)[/tex]
1. Start by isolating the term with [tex]\(x\)[/tex]:
[tex]\[ \frac{x-4}{3} = 5 \][/tex]
2. Multiply both sides by 3 to clear the fraction:
[tex]\[ x - 4 = 15 \][/tex]
3. Add 4 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[ x = 19 \][/tex]
So, [tex]\( x = 19 \)[/tex].
### Equation 8: [tex]\( x - \frac{x+2}{12} = \frac{5x}{2} \)[/tex]
1. Start by clearing the fraction by multiplying every term by 12:
[tex]\[ 12x - (x + 2) = 6 \times 5x \][/tex]
2. Simplify the equation:
[tex]\[ 12x - x - 2 = 30x \][/tex]
[tex]\[ 11x - 2 = 30x \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ 11x - 30x = 2 \][/tex]
[tex]\[ -19x = 2 \][/tex]
4. Divide by -19 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = -\frac{2}{19} \][/tex]
So, [tex]\( x = -\frac{2}{19} \)[/tex].
### Equation 9: [tex]\( x - \frac{5x-1}{3} = 4x - \frac{3}{5} \)[/tex]
1. Clear the fractions by multiplying every term by 15 (LCM of 3 and 5):
[tex]\[ 15x - 5(5x - 1) = 60x - 3 \][/tex]
2. Expand and simplify:
[tex]\[ 15x - 25x + 5 = 60x - 3 \][/tex]
[tex]\[ -10x + 5 = 60x - 3 \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ -10x - 60x = -3 - 5 \][/tex]
[tex]\[ -70x = -8 \][/tex]
4. Divide by -70 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{-8}{-70} = \frac{8}{70} = \frac{4}{35} \approx 0.114 \][/tex]
So, [tex]\( x \approx 0.200 \)[/tex]. (Correction: It's exact [tex]\(0.200\)[/tex])
### Equation 10: [tex]\( 10x - \frac{8x - 3}{4} = 2(x - 3) \)[/tex]
1. Clear the fraction by multiplying every term by 4:
[tex]\[ 40x - (8x - 3) = 8(x - 3) \][/tex]
2. Distribute and simplify:
[tex]\[ 40x - 8x + 3 = 8x - 24 \][/tex]
[tex]\[ 32x + 3 = 8x - 24 \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ 32x - 8x = -24 - 3 \][/tex]
[tex]\[ 24x = -27 \][/tex]
4. Divide by 24 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = -\frac{27}{24} = -\frac{9}{8} \][/tex]
So, [tex]\( x = -\frac{9}{8} \)[/tex].
### Equation 11: [tex]\(\frac{x-2}{3} - \frac{x-3}{4} = \frac{x-4}{5} \)[/tex]
1. Clear the fractions by multiplying every term by 60 (LCM of 3, 4, and 5):
[tex]\[ 20(x - 2) - 15(x - 3) = 12(x - 4) \][/tex]
2. Distribute and simplify:
[tex]\[ 20x - 40 - 15x + 45 = 12x - 48 \][/tex]
[tex]\[ 5x + 5 = 12x - 48 \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ 5x - 12x = -48 - 5 \][/tex]
[tex]\[ -7x = -53 \][/tex]
4. Divide by -7 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{53}{7} \][/tex]
So, [tex]\( x = \frac{53}{7} \)[/tex].
### Equation 12: [tex]\(\frac{x-1}{2} - \frac{x-2}{3} - \frac{x-3}{4} = -\frac{x-5}{5} \)[/tex]
1. Clear the fractions by multiplying every term by 60 (LCM of 2, 3, 4, and 5):
[tex]\[ 30(x - 1) - 20(x - 2) - 15(x - 3) = -12(x - 5) \][/tex]
2. Distribute and simplify:
[tex]\[ 30x - 30 - 20x + 40 - 15x + 45 = -12x + 60 \][/tex]
[tex]\[ (30x - 20x - 15x) = (-12x + 60 + 30 - 40 - 45) \][/tex]
[tex]\[ -5x - 30 + 40 + 45 = -12x + 60 \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ -5x + 12x = -30 + 40 + 45 - 60 \][/tex]
[tex]\[ 7x = 5 \][/tex]
4. Divide by 7 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{5}{7} \][/tex]
So, [tex]\( x = \frac{5}{7} \)[/tex].
### Equation 13: [tex]\( x - (5x - 1) - \frac{7 - 5x}{10} = 1 \)[/tex]
1. Clear the fraction by multiplying every term by 10:
[tex]\[ 10x - 10(5x - 1) - (7 - 5x) = 10 \][/tex]
2. Distribute and simplify:
[tex]\[ 10x - 50x + 10 - 7 + 5x = 10 \][/tex]
[tex]\[ (10x - 50x + 5x) + 10 - 7 = 10 \][/tex]
[tex]\[ -35x + 10 -7 = 10 \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ -35x + 3 = 10 \][/tex]
[tex]\[ -35x = 7 \][/tex]
4. Divide by -35 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = -\frac{7}{35} = -\frac{1}{5} \][/tex]
So, [tex]\( x = -\frac{1}{5} \)[/tex].
### Equation 14: [tex]\( 2x - \frac{5x - 6}{4} + \frac{1}{3}(x - 5) = -5x \)[/tex]
1. Clear the fraction by multiplying every term by 12 (LCM of 4 and 3):
[tex]\[ 24x - 3(5x - 6) + 4(x - 5) = -60x \][/tex]
2. Distribute and simplify:
[tex]\[ 24x - 15x + 18 + 4x - 20 = -60x \][/tex]
[tex]\[ (24x - 15x + 4x) + (18 - 20) = -60x \][/tex]
[tex]\[ 13x - 2 = -60x \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ 13x + 60x = 2 \][/tex]
[tex]\[ 73x = 73 \][/tex]
4. Divide by 73 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{2}{73} \][/tex]
So, [tex]\( x = \approx 0.0273972602739726 \)[/tex] (Exact: not fraction)
### Equation 15: [tex]\( 4 - \frac{10x + 1}{6} = 4x - \frac{16x + 3}{4} \)[/tex]
1. Clear the fraction by multiplying every term by 12 (LCM of 6 and 4):
[tex]\[ 48 - 2(10x + 1) = 12 * 4x - 3(16x + 3) \][/tex]
2. Distribute and simplify:
[tex]\[ 48 - 20x - 2 = 48x - 48x - 9 \][/tex]
[tex]\[ 46 - 20x = 48x - 9 \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ 46 = 48x - 9 + 20x \][/tex]
[tex]\[ 75= 48x + 20 x = 53 \][/tex]
\]
4. Divide by 4 to solve for [tex]\(x\)[/tex]:
\[
x = \frac{11}{4} = develop exact . calculation
]
So, \( x = 11/4).
### Equation 7: [tex]\(\frac{x-4}{3} - 5 = 0\)[/tex]
1. Start by isolating the term with [tex]\(x\)[/tex]:
[tex]\[ \frac{x-4}{3} = 5 \][/tex]
2. Multiply both sides by 3 to clear the fraction:
[tex]\[ x - 4 = 15 \][/tex]
3. Add 4 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[ x = 19 \][/tex]
So, [tex]\( x = 19 \)[/tex].
### Equation 8: [tex]\( x - \frac{x+2}{12} = \frac{5x}{2} \)[/tex]
1. Start by clearing the fraction by multiplying every term by 12:
[tex]\[ 12x - (x + 2) = 6 \times 5x \][/tex]
2. Simplify the equation:
[tex]\[ 12x - x - 2 = 30x \][/tex]
[tex]\[ 11x - 2 = 30x \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ 11x - 30x = 2 \][/tex]
[tex]\[ -19x = 2 \][/tex]
4. Divide by -19 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = -\frac{2}{19} \][/tex]
So, [tex]\( x = -\frac{2}{19} \)[/tex].
### Equation 9: [tex]\( x - \frac{5x-1}{3} = 4x - \frac{3}{5} \)[/tex]
1. Clear the fractions by multiplying every term by 15 (LCM of 3 and 5):
[tex]\[ 15x - 5(5x - 1) = 60x - 3 \][/tex]
2. Expand and simplify:
[tex]\[ 15x - 25x + 5 = 60x - 3 \][/tex]
[tex]\[ -10x + 5 = 60x - 3 \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ -10x - 60x = -3 - 5 \][/tex]
[tex]\[ -70x = -8 \][/tex]
4. Divide by -70 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{-8}{-70} = \frac{8}{70} = \frac{4}{35} \approx 0.114 \][/tex]
So, [tex]\( x \approx 0.200 \)[/tex]. (Correction: It's exact [tex]\(0.200\)[/tex])
### Equation 10: [tex]\( 10x - \frac{8x - 3}{4} = 2(x - 3) \)[/tex]
1. Clear the fraction by multiplying every term by 4:
[tex]\[ 40x - (8x - 3) = 8(x - 3) \][/tex]
2. Distribute and simplify:
[tex]\[ 40x - 8x + 3 = 8x - 24 \][/tex]
[tex]\[ 32x + 3 = 8x - 24 \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ 32x - 8x = -24 - 3 \][/tex]
[tex]\[ 24x = -27 \][/tex]
4. Divide by 24 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = -\frac{27}{24} = -\frac{9}{8} \][/tex]
So, [tex]\( x = -\frac{9}{8} \)[/tex].
### Equation 11: [tex]\(\frac{x-2}{3} - \frac{x-3}{4} = \frac{x-4}{5} \)[/tex]
1. Clear the fractions by multiplying every term by 60 (LCM of 3, 4, and 5):
[tex]\[ 20(x - 2) - 15(x - 3) = 12(x - 4) \][/tex]
2. Distribute and simplify:
[tex]\[ 20x - 40 - 15x + 45 = 12x - 48 \][/tex]
[tex]\[ 5x + 5 = 12x - 48 \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ 5x - 12x = -48 - 5 \][/tex]
[tex]\[ -7x = -53 \][/tex]
4. Divide by -7 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{53}{7} \][/tex]
So, [tex]\( x = \frac{53}{7} \)[/tex].
### Equation 12: [tex]\(\frac{x-1}{2} - \frac{x-2}{3} - \frac{x-3}{4} = -\frac{x-5}{5} \)[/tex]
1. Clear the fractions by multiplying every term by 60 (LCM of 2, 3, 4, and 5):
[tex]\[ 30(x - 1) - 20(x - 2) - 15(x - 3) = -12(x - 5) \][/tex]
2. Distribute and simplify:
[tex]\[ 30x - 30 - 20x + 40 - 15x + 45 = -12x + 60 \][/tex]
[tex]\[ (30x - 20x - 15x) = (-12x + 60 + 30 - 40 - 45) \][/tex]
[tex]\[ -5x - 30 + 40 + 45 = -12x + 60 \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ -5x + 12x = -30 + 40 + 45 - 60 \][/tex]
[tex]\[ 7x = 5 \][/tex]
4. Divide by 7 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{5}{7} \][/tex]
So, [tex]\( x = \frac{5}{7} \)[/tex].
### Equation 13: [tex]\( x - (5x - 1) - \frac{7 - 5x}{10} = 1 \)[/tex]
1. Clear the fraction by multiplying every term by 10:
[tex]\[ 10x - 10(5x - 1) - (7 - 5x) = 10 \][/tex]
2. Distribute and simplify:
[tex]\[ 10x - 50x + 10 - 7 + 5x = 10 \][/tex]
[tex]\[ (10x - 50x + 5x) + 10 - 7 = 10 \][/tex]
[tex]\[ -35x + 10 -7 = 10 \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ -35x + 3 = 10 \][/tex]
[tex]\[ -35x = 7 \][/tex]
4. Divide by -35 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = -\frac{7}{35} = -\frac{1}{5} \][/tex]
So, [tex]\( x = -\frac{1}{5} \)[/tex].
### Equation 14: [tex]\( 2x - \frac{5x - 6}{4} + \frac{1}{3}(x - 5) = -5x \)[/tex]
1. Clear the fraction by multiplying every term by 12 (LCM of 4 and 3):
[tex]\[ 24x - 3(5x - 6) + 4(x - 5) = -60x \][/tex]
2. Distribute and simplify:
[tex]\[ 24x - 15x + 18 + 4x - 20 = -60x \][/tex]
[tex]\[ (24x - 15x + 4x) + (18 - 20) = -60x \][/tex]
[tex]\[ 13x - 2 = -60x \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ 13x + 60x = 2 \][/tex]
[tex]\[ 73x = 73 \][/tex]
4. Divide by 73 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{2}{73} \][/tex]
So, [tex]\( x = \approx 0.0273972602739726 \)[/tex] (Exact: not fraction)
### Equation 15: [tex]\( 4 - \frac{10x + 1}{6} = 4x - \frac{16x + 3}{4} \)[/tex]
1. Clear the fraction by multiplying every term by 12 (LCM of 6 and 4):
[tex]\[ 48 - 2(10x + 1) = 12 * 4x - 3(16x + 3) \][/tex]
2. Distribute and simplify:
[tex]\[ 48 - 20x - 2 = 48x - 48x - 9 \][/tex]
[tex]\[ 46 - 20x = 48x - 9 \][/tex]
3. Get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ 46 = 48x - 9 + 20x \][/tex]
[tex]\[ 75= 48x + 20 x = 53 \][/tex]
\]
4. Divide by 4 to solve for [tex]\(x\)[/tex]:
\[
x = \frac{11}{4} = develop exact . calculation
]
So, \( x = 11/4).
