Question

solve for m.
\\(\frac{-1}{m + 15} = \frac{m}{50}\\)

Explanation:

Step1: Cross - multiply the fractions

To solve the equation \(\frac{-1}{m + 15}=\frac{m}{50}\), we can use the cross - multiplication property of proportions. If \(\frac{a}{b}=\frac{c}{d}\), then \(a\times d=b\times c\). So, we have \(- 1\times50=m\times(m + 15)\).
\(-50=m^{2}+15m\)

Step2: Rearrange into standard quadratic form

We can rewrite the equation from step 1 into the standard quadratic form \(ax^{2}+bx + c = 0\) (here our variable is \(m\)). So, we get \(m^{2}+15m+50 = 0\).

Step3: Factor the quadratic equation

We need to factor the quadratic \(m^{2}+15m + 50\). We look for two numbers that multiply to \(50\) and add up to \(15\). The numbers \(5\) and \(10\) satisfy this condition since \(5\times10 = 50\) and \(5 + 10=15\). So, the factored form is \((m + 5)(m+10)=0\).

Step4: Solve for m

Using the zero - product property, if \((m + 5)(m + 10)=0\), then either \(m+5 = 0\) or \(m + 10=0\).

• If \(m+5=0\), then \(m=-5\).
• If \(m + 10=0\), then \(m=-10\).

We need to check these solutions in the original equation because we have a rational equation (there is a variable in the denominator).

Check \(m=-5\):

Left - hand side: \(\frac{-1}{-5 + 15}=\frac{-1}{10}=-0.1\)
Right - hand side: \(\frac{-5}{50}=-0.1\)
So, \(m = - 5\) is a valid solution.

Check \(m=-10\):

Left - hand side: \(\frac{-1}{-10 + 15}=\frac{-1}{5}=-0.2\)
Right - hand side: \(\frac{-10}{50}=-0.2\)
So, \(m=-10\) is also a valid solution.

Answer:

\(m=-5\) or \(m = - 10\)