Question

How do you solve $3x=13+5y$ and $4x-7y-17=0$?

Answer 1

Hint: In this problem we need to solve the given two pairs of linear equations in two variables. We have the given first equation as $3x=13+5y$. From this equation we can have the value of $x$ by dividing the whole equation with $3$ on both sides of the equation. Now we will substitute this value in the given second equation which is $4x-7y-17=0$ and simplify the equation by performing the operations like fractions additions, fractions subtraction etc. After all the operations are done to simplify the equation, we will get the value of $y$. From the value of $y$, we will calculate the value of $x$ by substituting the value of $y$ in any one of the equations we have.

Complete step-by-step solution:
Given equations, $3x=13+5y$ and $4x-7y-17=0$.
Considering the first given equation which is $3x=13+5y$. Dividing the above equation with $3$ on both sides, then we will get
$\begin{align}
  & \Rightarrow \dfrac{3x}{3}=\dfrac{13+5y}{3} \\
 & \Rightarrow x=\dfrac{13+5y}{3}....\left( \text{i} \right) \\
\end{align}$
Substituting the above obtained $x$ value in second given equation which is $4x-7y-17=0$, then we will get
$\Rightarrow 4\left( \dfrac{13+5y}{3} \right)-7y-17=0$
Simplifying the above equation by taking LCM on both sides, then we will get
$\Rightarrow \dfrac{4\left( 13+5y \right)-3\left( 7y \right)-3\left( 17 \right)}{3}=0$
Using distribution law of multiplication in the above equation, then we will get
$\begin{align}
  & \Rightarrow 52+20y-21y-51=0 \\
 & \Rightarrow -y+1=0 \\
\end{align}$
Adding $y$ on both sides of the above equation, then we will have
$\begin{align}
  & \Rightarrow -y+1+y=y \\
 & \Rightarrow y=1 \\
\end{align}$
Now substituting the above $y$ value in $\left( \text{i} \right)$, then we will get
$\begin{align}
  & x=\dfrac{13+5\left( 1 \right)}{3} \\
 & \Rightarrow x=\dfrac{18}{3} \\
 & \Rightarrow x=6 \\
\end{align}$
Hence the solution of the given system $3x=13+5y$ and $4x-7y-17=0$ is $\left( x,y \right)=\left( 6,1 \right)$.

Note: We can also use the graph method to solve the given system. We will graph both the equations we have in the problem and note the point of intersection as the solution of the system. When we plot the graph of the given system $3x=13+5y$ and $4x-7y-17=0$, then we will get

From the above graph also, we can observe the result as $\left( 6,1 \right)$.