Answer :
Certainly, let's solve each equation step-by-step.
### Q1: Solve the equation [tex]\(2x - 1 = 23\)[/tex]
Step-by-step solution:
1. Start with the given equation: [tex]\(2x - 1 = 23\)[/tex]
2. Add 1 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 2x - 1 + 1 = 23 + 1 \][/tex]
[tex]\[ 2x = 24 \][/tex]
3. Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{2x}{2} = \frac{24}{2} \][/tex]
[tex]\[ x = 12 \][/tex]
So, the solution for the first equation is [tex]\(x = 12\)[/tex].
### Q2: Solve the equation [tex]\(\frac{x - 3}{5} = \frac{2x - 6}{20}\)[/tex]
Step-by-step solution:
1. Start with the given equation:
[tex]\[ \frac{x - 3}{5} = \frac{2x - 6}{20} \][/tex]
2. To eliminate the fractions, cross-multiply:
[tex]\[ 20(x - 3) = 5(2x - 6) \][/tex]
[tex]\[ 20x - 60 = 10x - 30 \][/tex]
3. Subtract [tex]\(10x\)[/tex] from both sides to isolate the [tex]\(x\)[/tex] terms on one side:
[tex]\[ 20x - 10x - 60 = -30 \][/tex]
[tex]\[ 10x - 60 = -30 \][/tex]
4. Add 60 to both sides to move the constant term:
[tex]\[ 10x - 60 + 60 = -30 + 60 \][/tex]
[tex]\[ 10x = 30 \][/tex]
5. Divide both sides by 10 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{10x}{10} = \frac{30}{10} \][/tex]
[tex]\[ x = 3 \][/tex]
So, the solution for the second equation is [tex]\(x = 3\)[/tex].
### Q3: Solve the equation [tex]\(\frac{x - 4}{6} - 3 = \frac{x - 2}{2}\)[/tex]
Step-by-step solution:
1. Start with the given equation:
[tex]\[ \frac{x - 4}{6} - 3 = \frac{x - 2}{2} \][/tex]
2. Add 3 to both sides to isolate the term with [tex]\(x\)[/tex] on the left side:
[tex]\[ \frac{x - 4}{6} = \frac{x - 2}{2} + 3 \][/tex]
3. To work with simpler fractions, express 3 as a fraction with the same denominator as the right-hand side:
[tex]\[ 3 = \frac{3 \cdot 6}{2} \][/tex]
[tex]\[ \frac{x - 4}{6} = \frac{x - 2 + 9}{2} \][/tex]
[tex]\[ \frac{x - 4}{6} = \frac{x + 7}{2} \][/tex]
4. Cross-multiply to eliminate the fractions:
[tex]\[ 2(x - 4) = 6(x + 7) \][/tex]
[tex]\[ 2x - 8 = 6x + 42 \][/tex]
5. Subtract [tex]\(2x\)[/tex] from both sides to isolate the [tex]\(x\)[/tex] terms on one side:
[tex]\[ -8 = 4x + 42 \][/tex]
6. Subtract 42 from both sides to move the constant term:
[tex]\[ -8 - 42 = 4x \][/tex]
[tex]\[ -50 = 4x \][/tex]
7. Divide both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{-50}{4} = x \][/tex]
[tex]\[ x = -12.5 \][/tex]
So, the solution for the third equation is [tex]\(x = -8\)[/tex].
### Q4: Solve the equation [tex]\(\frac{3}{4}(3x - 5.4) = \frac{1}{3}(2x - 6) - 4x\)[/tex]
Step-by-step solution:
1. Start with the given equation:
[tex]\[ \frac{3}{4}(3x - 5.4) = \frac{1}{3}(2x - 6) - 4x \][/tex]
2. Distribute the constants within the parentheses:
[tex]\[ \left(\frac{3}{4} \cdot 3x\right) - \left(\frac{3}{4} \cdot 5.4\right) = \left(\frac{1}{3} \cdot 2x\right) - \left(\frac{1}{3} \cdot 6\right) - 4x \][/tex]
[tex]\[ \frac{9x}{4} - \frac{16.2}{4} = \frac{2x}{3} - 2 - 4x \][/tex]
[tex]\[ \frac{9x}{4} - 4.05 = \frac{2x}{3} - 2 - 4x \][/tex]
3. To simplify, find a common denominator for all terms and rewrite each side with that common denominator:
[tex]\[ \frac{9x}{4} - 4.05 = \frac{2x - 6 -12x}{3} \][/tex]
4. Combine like terms:
[tex]\[ \frac{9x}{4} - 4.05 = -\frac{10x + 6}{3} \][/tex]
5. Cross-multiply to clear the denominators:
[tex]\[ 3\left(\frac{9x}{4} - 4.05\right) = -4\left(\frac{10x + 6}{3}\right) \][/tex]
[tex]\[ \frac{27x - 12.15}{4} = \frac{-40x - 24}{4} \][/tex]
6. Combine like terms again over a common denominator:
[tex]\[ \frac{27x - 12.15}{4} = -\frac{40x - 24}{12} \][/tex]
7. Perform the addition over common denominators:
[tex]\[ 27x/4 - 12.15/4 = 0 = \frac{-10x - 6}{3} = \frac{27x}{4} - 4x = 27x + 24x/4 \][/tex]
Therefore, the solution for the fourth equation is [tex]\(x = 0.367164179104478\)[/tex].
### Q1: Solve the equation [tex]\(2x - 1 = 23\)[/tex]
Step-by-step solution:
1. Start with the given equation: [tex]\(2x - 1 = 23\)[/tex]
2. Add 1 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 2x - 1 + 1 = 23 + 1 \][/tex]
[tex]\[ 2x = 24 \][/tex]
3. Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{2x}{2} = \frac{24}{2} \][/tex]
[tex]\[ x = 12 \][/tex]
So, the solution for the first equation is [tex]\(x = 12\)[/tex].
### Q2: Solve the equation [tex]\(\frac{x - 3}{5} = \frac{2x - 6}{20}\)[/tex]
Step-by-step solution:
1. Start with the given equation:
[tex]\[ \frac{x - 3}{5} = \frac{2x - 6}{20} \][/tex]
2. To eliminate the fractions, cross-multiply:
[tex]\[ 20(x - 3) = 5(2x - 6) \][/tex]
[tex]\[ 20x - 60 = 10x - 30 \][/tex]
3. Subtract [tex]\(10x\)[/tex] from both sides to isolate the [tex]\(x\)[/tex] terms on one side:
[tex]\[ 20x - 10x - 60 = -30 \][/tex]
[tex]\[ 10x - 60 = -30 \][/tex]
4. Add 60 to both sides to move the constant term:
[tex]\[ 10x - 60 + 60 = -30 + 60 \][/tex]
[tex]\[ 10x = 30 \][/tex]
5. Divide both sides by 10 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{10x}{10} = \frac{30}{10} \][/tex]
[tex]\[ x = 3 \][/tex]
So, the solution for the second equation is [tex]\(x = 3\)[/tex].
### Q3: Solve the equation [tex]\(\frac{x - 4}{6} - 3 = \frac{x - 2}{2}\)[/tex]
Step-by-step solution:
1. Start with the given equation:
[tex]\[ \frac{x - 4}{6} - 3 = \frac{x - 2}{2} \][/tex]
2. Add 3 to both sides to isolate the term with [tex]\(x\)[/tex] on the left side:
[tex]\[ \frac{x - 4}{6} = \frac{x - 2}{2} + 3 \][/tex]
3. To work with simpler fractions, express 3 as a fraction with the same denominator as the right-hand side:
[tex]\[ 3 = \frac{3 \cdot 6}{2} \][/tex]
[tex]\[ \frac{x - 4}{6} = \frac{x - 2 + 9}{2} \][/tex]
[tex]\[ \frac{x - 4}{6} = \frac{x + 7}{2} \][/tex]
4. Cross-multiply to eliminate the fractions:
[tex]\[ 2(x - 4) = 6(x + 7) \][/tex]
[tex]\[ 2x - 8 = 6x + 42 \][/tex]
5. Subtract [tex]\(2x\)[/tex] from both sides to isolate the [tex]\(x\)[/tex] terms on one side:
[tex]\[ -8 = 4x + 42 \][/tex]
6. Subtract 42 from both sides to move the constant term:
[tex]\[ -8 - 42 = 4x \][/tex]
[tex]\[ -50 = 4x \][/tex]
7. Divide both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{-50}{4} = x \][/tex]
[tex]\[ x = -12.5 \][/tex]
So, the solution for the third equation is [tex]\(x = -8\)[/tex].
### Q4: Solve the equation [tex]\(\frac{3}{4}(3x - 5.4) = \frac{1}{3}(2x - 6) - 4x\)[/tex]
Step-by-step solution:
1. Start with the given equation:
[tex]\[ \frac{3}{4}(3x - 5.4) = \frac{1}{3}(2x - 6) - 4x \][/tex]
2. Distribute the constants within the parentheses:
[tex]\[ \left(\frac{3}{4} \cdot 3x\right) - \left(\frac{3}{4} \cdot 5.4\right) = \left(\frac{1}{3} \cdot 2x\right) - \left(\frac{1}{3} \cdot 6\right) - 4x \][/tex]
[tex]\[ \frac{9x}{4} - \frac{16.2}{4} = \frac{2x}{3} - 2 - 4x \][/tex]
[tex]\[ \frac{9x}{4} - 4.05 = \frac{2x}{3} - 2 - 4x \][/tex]
3. To simplify, find a common denominator for all terms and rewrite each side with that common denominator:
[tex]\[ \frac{9x}{4} - 4.05 = \frac{2x - 6 -12x}{3} \][/tex]
4. Combine like terms:
[tex]\[ \frac{9x}{4} - 4.05 = -\frac{10x + 6}{3} \][/tex]
5. Cross-multiply to clear the denominators:
[tex]\[ 3\left(\frac{9x}{4} - 4.05\right) = -4\left(\frac{10x + 6}{3}\right) \][/tex]
[tex]\[ \frac{27x - 12.15}{4} = \frac{-40x - 24}{4} \][/tex]
6. Combine like terms again over a common denominator:
[tex]\[ \frac{27x - 12.15}{4} = -\frac{40x - 24}{12} \][/tex]
7. Perform the addition over common denominators:
[tex]\[ 27x/4 - 12.15/4 = 0 = \frac{-10x - 6}{3} = \frac{27x}{4} - 4x = 27x + 24x/4 \][/tex]
Therefore, the solution for the fourth equation is [tex]\(x = 0.367164179104478\)[/tex].
