Question

1. \\(\frac{5}{m} = \frac{8}{2m - 6}\\)

Explanation:

Step1: Cross - multiply the fractions

To solve the equation \(\frac{5}{m}=\frac{8}{2m - 6}\), we cross - multiply. Cross - multiplying gives us \(5\times(2m - 6)=8\times m\).

Step2: Expand the left - hand side

Using the distributive property \(a(b - c)=ab - ac\), where \(a = 5\), \(b = 2m\) and \(c = 6\), we get \(5\times2m-5\times6 = 8m\), which simplifies to \(10m-30 = 8m\).

Step3: Isolate the variable \(m\)

Subtract \(8m\) from both sides of the equation: \(10m-8m - 30=8m - 8m\), which gives \(2m-30 = 0\). Then add 30 to both sides: \(2m-30 + 30=0 + 30\), so \(2m=30\).

Step4: Solve for \(m\)

Divide both sides of the equation \(2m = 30\) by 2: \(m=\frac{30}{2}=15\). We should also check if this solution makes the original denominators non - zero. For the denominator \(m\), when \(m = 15\), \(m=15
eq0\). For the denominator \(2m - 6\), when \(m = 15\), \(2\times15-6=30 - 6 = 24
eq0\). So \(m = 15\) is a valid solution.

Answer:

\(m = 15\)