Let's solve the equation step-by-step while applying the appropriate mathematical properties. The original equation is:
[tex]\[
\frac{17}{3} - \frac{3}{4}x = \frac{1}{2}x + 5
\][/tex]
Step 1: Start with the given equation.
[tex]\[
\frac{17}{3} - \frac{3}{4}x = \frac{1}{2}x + 5
\][/tex]
Justification: The original equation given to us.
Step 2: Apply the Subtraction Property of Equality to subtract [tex]\(\frac{17}{3}\)[/tex] from both sides:
[tex]\[
\frac{17}{3} - \frac{3}{4}x - \frac{17}{3} = \frac{1}{2}x + 5 - \frac{17}{3}
\][/tex]
Justification: Subtract the same value ([tex]\(\frac{17}{3}\)[/tex]) from both sides to keep the equation balanced.
Step 3: Simplify both sides:
[tex]\[
-\frac{3}{4}x = \frac{1}{2}x + 5 - \frac{17}{3}
\][/tex]
Justification: Simplify the left-hand side by cancelling out [tex]\(\frac{17}{3}\)[/tex].
Next, convert the constant on the right-hand side to a common denominator:
[tex]\[
5 - \frac{17}{3} = \frac{15}{3} - \frac{17}{3} = -\frac{2}{3}
\][/tex]
So,
[tex]\[
-\frac{3}{4}x = \frac{1}{2}x - \frac{2}{3}
\][/tex]
Justification: Simplify the arithmetic to combine terms on the right-hand side.
Step 4: Apply the Subtraction Property of Equality to subtract [tex]\(\frac{1}{2}x\)[/tex] from both sides:
[tex]\[
-\frac{3}{4}x - \frac{1}{2}x = \frac{1}{2}x - \frac{2}{3} - \frac{1}{2}x
\][/tex]
Justification: Subtract [tex]\(\frac{1}{2}x\)[/tex] from both sides to isolate [tex]\(x\)[/tex] terms on the left-hand side.
Step 5: Simplify both sides:
[tex]\[
-\frac{3}{4}x - \frac{1}{2}x = -\frac{2}{3}
\][/tex]
Combine like terms on the left-hand side.
To do this, find a common denominator for the coefficients of [tex]\(x\)[/tex]:
[tex]\[
-\frac{3}{4}x - \frac{1}{2}x = -\frac{3}{4}x - \frac{2}{4}x = -\frac{5}{4}x
\][/tex]
So,
[tex]\[
-\frac{5}{4}x = -\frac{2}{3}
\][/tex]
Step 6: Apply the Multiplication Property of Equality to isolate [tex]\(x\)[/tex]. Multiply both sides by [tex]\(-\frac{4}{5}\)[/tex]:
[tex]\[
-\frac{5}{4}x \cdot -\frac{4}{5} = -\frac{2}{3} \cdot -\frac{4}{5}
\][/tex]
Justification: To solve for [tex]\(x\)[/tex], multiply both sides by the reciprocal of [tex]\(-\frac{5}{4}\)[/tex], which is [tex]\(-\frac{4}{5}\)[/tex].
Step 7: Simplify both sides:
[tex]\[
x = \frac{8}{15}
\][/tex]
Justification: Simplify the right side [tex]\((-\frac{2}{3} \cdot -\frac{4}{5} = \frac{8}{15})\)[/tex].
Therefore, the solution to the equation is:
[tex]\[
x = \frac{8}{15}
\][/tex]
Each step aligns with a mathematical principle to ensure the correctness of the solution.
