Question

solve for r. separate multiple solutions with a comma. if there are no solutions, enter
one\. \\(\frac{r}{r - 45} = \frac{5}{2}\\)

Explanation:

Step1: Cross - multiply the equation

To solve the equation \(\frac{r}{r - 45}=\frac{5}{2}\), we can cross - multiply. Cross - multiplying gives us \(2\times r=5\times(r - 45)\).

Step2: Expand the right - hand side

Expanding the right - hand side of the equation \(2r = 5(r-45)\), we use the distributive property \(a(b + c)=ab+ac\) (here \(a = 5\), \(b=r\), \(c=- 45\)). So \(2r=5r-225\).

Step3: Isolate the variable \(r\)

Subtract \(2r\) from both sides: \(0 = 5r-2r-225\), which simplifies to \(0 = 3r-225\). Then add 225 to both sides: \(3r=225\).

Step4: Solve for \(r\)

Divide both sides by 3: \(r=\frac{225}{3}=75\).
We need to check if this solution makes the original denominator non - zero. When \(r = 75\), the denominator \(r - 45=75 - 45 = 30
eq0\), so \(r = 75\) is a valid solution.

Answer:

75