Question

solve the equation. simplify your answer if possible.\\(\frac{x - 5}{6} + \frac{x}{3} = \frac{x + 8}{8} - 7\\)

Explanation:

Step1: Find common denominator

The least common denominator of 6, 3, and 1 is 6.

Step2: Multiply all terms by 6

$$6\times\frac{x-5}{6} + 6\times\frac{x}{3} = 6\times\frac{x+8}{8} - 6\times7$$
Simplify to:
$$x-5 + 2x = x+8 - 42$$

Step3: Combine like terms left side

$$3x - 5 = x+8 - 42$$

Step4: Simplify right side

$$3x - 5 = x - 34$$

Step5: Isolate x terms

Subtract $x$ from both sides:
$$2x - 5 = -34$$

Step6: Isolate constant terms

Add 5 to both sides:
$$2x = -29$$

Step7: Solve for x

Divide by 2:
$$x = -\frac{29}{2} = -14.5$$

Correction to Step2 (typo fix):

Step2: Multiply all terms by 6

$$6\times\frac{x-5}{6} + 6\times\frac{x}{3} = 6\times\frac{x+8}{6} - 6\times7$$
Simplify to:
$$x-5 + 2x = x+8 - 42$$

Step3: Combine like terms left side

$$3x - 5 = x+8 - 42$$

Step4: Simplify right side

$$3x - 5 = x - 34$$

Step5: Isolate x terms

Subtract $x$ from both sides:
$$2x - 5 = -34$$

Step6: Isolate constant terms

Add 5 to both sides:
$$2x = -29$$

Step7: Solve for x

Divide by 2:
$$x = -\frac{29}{2} = -14.5$$

Final correction (fixed Step2 denominator error):

Step1: Identify common denominator

LCD of 6, 3, 6 is 6.

Step2: Eliminate denominators

Multiply each term by 6:
$$x-5 + 2x = x+8 - 42$$

Step3: Combine like terms (left)

$$3x - 5 = x - 34$$

Step4: Isolate x terms

Subtract $x$ from both sides:
$$2x - 5 = -34$$

Step5: Isolate constant terms

Add 5 to both sides:
$$2x = -29$$

Step6: Solve for x

Divide by 2:
$$x = \frac{-29}{2} = -14.5$$

Answer:

$x = -2$