Question
Solve for \( r \) \( -r+3-\frac{1}{9} r=2 r+1-\frac{8}{9} r \) \( r= \)
Ask by Maxwell Olson. in the United States
Jan 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( r = \frac{9}{10} \)
Solution
To solve the equation for \( r \):
\[
-r + 3 - \frac{1}{9}r = 2r + 1 - \frac{8}{9}r
\]
**Step 1: Combine like terms**
**Left Side:**
\[
-r - \frac{1}{9}r + 3 = -\left(\frac{9}{9} + \frac{1}{9}\right)r + 3 = -\frac{10}{9}r + 3
\]
**Right Side:**
\[
2r - \frac{8}{9}r + 1 = \left(\frac{18}{9} - \frac{8}{9}\right)r + 1 = \frac{10}{9}r + 1
\]
**Step 2: Set the simplified expressions equal to each other**
\[
-\frac{10}{9}r + 3 = \frac{10}{9}r + 1
\]
**Step 3: Move all \( r \)-terms to one side and constants to the other**
\[
-\frac{10}{9}r - \frac{10}{9}r = 1 - 3
\]
\[
-\frac{20}{9}r = -2
\]
**Step 4: Solve for \( r \)**
\[
r = \left(-2\right) \div \left(-\frac{20}{9}\right) = \frac{18}{20} = \frac{9}{10}
\]
**Final Answer:**
\[
r = \frac{9}{10}
\]
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Simplify this solution Bonus Knowledge
To solve the equation \( -r+3-\frac{1}{9} r=2 r+1-\frac{8}{9} r \), we first simplify both sides. Combining the terms on the left gives us: \[ -r - \frac{1}{9} r + 3 = -\frac{10}{9} r + 3 \] On the right, combine the \( r \) terms: \[ 2 r - \frac{8}{9} r + 1 = \frac{18}{9} r - \frac{8}{9} r + 1 = \frac{10}{9} r + 1 \] Now we have: \[ -\frac{10}{9} r + 3 = \frac{10}{9} r + 1 \] To eliminate the fractions, multiply through by 9: \[ -10r + 27 = 10r + 9 \] Now, move the \( r \) terms to one side and constants to the other: \[ 27 - 9 = 10r + 10r \implies 18 = 20r \] Now, solve for \( r \): \[ r = \frac{18}{20} = \frac{9}{10} \] Thus, the solution is: \[ r = \frac{9}{10} \]
