Question

solve the equation and check your solutic
$-\frac{1}{9}(x - 27) + \frac{1}{3}(x + 3) = x + 11$
$x = \square$

Explanation:

Step1: Distribute the fractions

First, we distribute the \(-\frac{1}{9}\) and \(\frac{1}{3}\) into the parentheses:

$$ -\frac{1}{9}x + \frac{27}{9} + \frac{1}{3}x + \frac{3}{3} = x + 11 $$

Simplify the constants: \(\frac{27}{9} = 3\) and \(\frac{3}{3} = 1\), so the equation becomes:

$$ -\frac{1}{9}x + 3 + \frac{1}{3}x + 1 = x + 11 $$

Step2: Combine like terms

Combine the \(x\)-terms and the constant terms on the left side:
For the \(x\)-terms: \(-\frac{1}{9}x + \frac{1}{3}x = -\frac{1}{9}x + \frac{3}{9}x = \frac{2}{9}x\)
For the constants: \(3 + 1 = 4\)
So the equation is now:

$$ \frac{2}{9}x + 4 = x + 11 $$

Step3: Isolate the variable \(x\)

Subtract \(\frac{2}{9}x\) from both sides:

$$ 4 = x - \frac{2}{9}x + 11 $$

Simplify the \(x\)-terms on the right: \(x - \frac{2}{9}x = \frac{9}{9}x - \frac{2}{9}x = \frac{7}{9}x\)
So we have:

$$ 4 = \frac{7}{9}x + 11 $$

Now subtract 11 from both sides:

$$ 4 - 11 = \frac{7}{9}x $$

$$ -7 = \frac{7}{9}x $$

Step4: Solve for \(x\)

Multiply both sides by \(\frac{9}{7}\) to solve for \(x\):

$$ x = -7 \times \frac{9}{7} $$

The 7s cancel out, leaving:

$$ x = -9 $$

Step5: Check the solution

Substitute \(x = -9\) back into the original equation:
Left side: \(-\frac{1}{9}(-9 - 27) + \frac{1}{3}(-9 + 3)\)
First, calculate inside the parentheses: \(-9 - 27 = -36\) and \(-9 + 3 = -6\)
Then: \(-\frac{1}{9}(-36) + \frac{1}{3}(-6) = 4 - 2 = 2\)
Right side: \(-9 + 11 = 2\)
Since left side = right side, the solution is correct.

Answer:

\(x = -9\)