Sure, let's solve these equations one by one.
Equation 3:
[tex]$\frac{x}{6} + 5 = \frac{2}{3} + x$[/tex]
To solve this equation, follow these steps:
1. Get rid of the fractions by multiplying every term by the common denominator, which is 6 in this case:
[tex]$x + 30 = 4 + 6x$[/tex]
2. Rearrange the equation to gather all terms involving [tex]\(x\)[/tex] on one side:
[tex]$x - 6x = 4 - 30$[/tex]
[tex]$-5x = -26$[/tex]
3. Divide both sides by -5 to isolate [tex]\(x\)[/tex]:
[tex]$x = \frac{-26}{-5}$[/tex]
[tex]$x = 5.2$[/tex]
So, the solution for Equation 3 is:
[tex]$x = 5.2$[/tex]
Equation 4:
[tex]$\frac{2}{3x} + \frac{1}{2} = \frac{3}{2x} + \frac{13}{6}$[/tex]
To solve this equation, follow these steps:
1. Get rid of the fractions by multiplying every term by the common denominator, which is [tex]\(6x\)[/tex] in this case:
[tex]$6x \left( \frac{2}{3x} + \frac{1}{2} \right) = 6x \left( \frac{3}{2x} + \frac{13}{6} \right)$[/tex]
2. Simplify the equation:
[tex]$4 + 3x = 9 + 13x$[/tex]
3. Rearrange the equation to gather all terms involving [tex]\(x\)[/tex] on one side:
[tex]$4 + 3x - 9 = 13x$[/tex]
[tex]$-5 = 10x$[/tex]
4. Divide both sides by 10 to isolate [tex]\(x\)[/tex]:
[tex]$x = \frac{-5}{10}$[/tex]
[tex]$x = -0.5$[/tex]
So, the solution for Equation 4 is:
[tex]$x = -0.5$[/tex]
Equation 5:
[tex]$\frac{6 - 8x}{10} = \frac{8 - 6x}{8} - \frac{10 - 4x}{4}$[/tex]
To solve this equation, follow these steps:
1. Combine the fractions on the right-hand side to a single fraction:
[tex]$\frac{6 - 8x}{10} = \frac{8 - 6x}{8} - \frac{10 - 4x}{4}$[/tex]
2. Find the common denominator for the fractions on the right-hand side, which is 8:
[tex]$\frac{8 - 6x}{8} - \frac{20 - 8x}{8}$[/tex]
[tex]$\frac{8 - 6x - (20 - 8x)}{8}$[/tex]
[tex]$\frac{8 - 6x - 20 + 8x}{8}$[/tex]
[tex]$\frac{-12 + 2x}{8}$[/tex]
[tex]$\frac{2x - 12}{8}$[/tex]
3. Now set the left-hand side equal to the right-hand side:
[tex]$\frac{6 - 8x}{10} = \frac{2x - 12}{8}$[/tex]
4. Cross-multiply to get rid of the fractions:
[tex]$(6 - 8x) \cdot 8 = (2x - 12) \cdot 10$[/tex]
5. Simplify the equation:
[tex]$48 - 64x = 20x - 120$[/tex]
6. Rearrange the equation to gather all terms involving [tex]\(x\)[/tex] on one side:
[tex]$48 + 120 = 20x + 64x$[/tex]
[tex]$168 = 84x$[/tex]
7. Divide both sides by 84 to isolate [tex]\(x\)[/tex]:
[tex]$x = \frac{168}{84}$[/tex]
[tex]$x = 2$[/tex]
So, the solution for Equation 5 is:
[tex]$x = 2$[/tex]
