Question

How do you solve $\dfrac{4}{m-8}=\dfrac{8}{2}$ ? \[\]

Answer 1

Hint: We recall the linear equations and basic properties of linear equations. We solve the given linear equation by eliminating the constant terms at the left hand side so that only the variable term of $m$ remains at one side. We simplify the right hand side , multiply $m-8$ both sides of the given equation , divide 4 both sides and add 8 both sides . \[\]

Complete step by step answer:
We know from algebra that the linear equation in one variable $x$ and constants $a\ne 0,b$is given by
\[ax=b\]
The term with which the variable is multiplied is called variable term and with whom not multiplied is called a constant term. We also know that if we add, subtract, multiply or divide the same number on both sides of the equation, equality holds. It is called balancing the equation. It means for some term $c$ we have,
\[\begin{align}
  & ax+c=b+c \\
 & ax-c=b-c \\
 & ax\times c=b\times c \\
 & \dfrac{ax}{c}=\dfrac{b}{c} \\
\end{align}\]
 When we are asked to solve for unknown variable in an equation it means we have to find the value or values of the unknown variable for which the equation satisfies. We are given the following linear equation
\[\dfrac{4}{m-8}=\dfrac{8}{2}\]
We see that the unknown variable is here $m$ and we have to find its value. So we eliminate the constants $4,8$ in the right hand side of the given equation. We begin by simplifying the right hand side $\left( 8\div 2 \right)$ to have
\[\Rightarrow \dfrac{4}{m-8}=4\]
 We multiply $m-8$ both sides of the above equation to have;
\[\begin{align}
  & \dfrac{4}{m-8}\left( m-8 \right)=4\left( m-8 \right) \\
 & \Rightarrow 4=4\left( m-8 \right) \\
\end{align}\]
We divide both sides of the above step by 4 t o have;
\[\begin{align}
  & \Rightarrow \dfrac{4}{4}=\dfrac{4\left( m-8 \right)}{4} \\
 & \Rightarrow 1=1\cdot \left( m-8 \right) \\
 & \Rightarrow 1=m-8 \\
\end{align}\]
We add 8 both sides of the above equation to have;
\[\begin{align}
  & \Rightarrow 1+8=m-8+8 \\
 & \Rightarrow 9=m \\
 & \Rightarrow m=9 \\
\end{align}\]

So the solution of the given equation is $m=9$.\[\]

Note: We note that sometimes solutions may not satisfy the original equation which are called extraneous solutions. If we would have obtained $m=8$ it would have been an extraneous solution. We can alternatively solve using cross-multiplication that is $\dfrac{a}{b}=\dfrac{c}{d}\Rightarrow ad=bc$. We cross-multiply the original equation to have;
\[\begin{align}
  & \dfrac{4}{m-8}=\dfrac{8}{2} \\
 & \Rightarrow 4\cdot 2=8\left( m-8 \right) \\
 & \Rightarrow 8=8\left( m-8 \right) \\
\end{align}\]
We divide both sides by 8 to have;
\[\begin{align}
  & \Rightarrow \dfrac{8}{8}=\dfrac{8\left( m-8 \right)}{8} \\
 & \Rightarrow 1=m-8 \\
\end{align}\]
We add 8 both sides to have;
\[\begin{align}
  & \Rightarrow 1+8=m-8+8 \\
 & \Rightarrow 9=m \\
 & \Rightarrow m=9 \\
\end{align}\]