Question

How do you solve the system of $y=3x+17$ and $y=-2x-8$?

Answer 1

Hint: To solve the given system of equations, we can use the method of substitution. Since y is explicitly expressed in terms of x in both of the given equations, we can substitute y from any of the two equations into the other equation and solve the resulting equation for x. Then on substituting the value of x in any of the two equations, we will get the value of y and hence the solution of the given system.

Complete step-by-step solution:
The given pair of the equations is
$\begin{align}
  & \Rightarrow y=3x+17.......\left( i \right) \\
 & \Rightarrow y=-2x-8.......\left( ii \right) \\
\end{align}$
Let us use the substitution method to solve the given system of equations. For this, let us substitute the value of y from the equation (ii) into the equation (i) to get
\[\Rightarrow -2x-8=3x+17\]
Adding $-2x$ both the sides, we get
\[\begin{align}
  & \Rightarrow -2x-8+2x=3x+17+2x \\
 & \Rightarrow -8=5x+17 \\
\end{align}\]
Now, on subtracting $17$ from both the sides we get
$\begin{align}
  & \Rightarrow -8-17=5x+17-17 \\
 & \Rightarrow -25=5x \\
 & \Rightarrow 5x=-25 \\
\end{align}$
Finally on dividing both sides by $5$, we get
$\begin{align}
  & \Rightarrow \dfrac{5x}{5}=-\dfrac{25}{5} \\
 & \Rightarrow x=-5 \\
\end{align}$
Now, we substitute this in the equation (i) to get
$\begin{align}
  & \Rightarrow y=3\left( -5 \right)+17 \\
 & \Rightarrow y=-15+17 \\
 & \Rightarrow y=2 \\
\end{align}$
Hence, we have finally obtained the solution to the given system of equations as $x=-5$ and $y=2$.

Note: We can also use the method of elimination to eliminate the variable y from the given equations and solve the given system. For this, we need to subtract one equation from the other. The other methods such as the cross multiplication method can also be used here. Further, we must check the final solution by substituting it back into the given equations.