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# How do you determine whether a linear system has one solution, many solutions, or no solution when given $2x+5y=-16$ and $6x+y=20$?

Last updated date: 22nd Feb 2024
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Hint: In this problem we need to determine about the solution for the given linear system. For this we will convert each equation into slope intercept form which is $y=mx+c$. For this we will apply several arithmetic operations according to the equation we have and simply the equation and write the values of slope values of both the equations. Now we will compare the slopes of both the equations, if we get the different values of slopes for both equations, then we have one solution for the given system. If we get the same slope for both the equations, then we will have zero solution. If we get both values of slope and intercept as the same then the system has infinite solution. From this we will get our required result.

Complete step by step solution:
Given equations are $2x+5y=-16$, $6x+y=20$.
Considering the first equation $2x+5y=-16$.
Shifting the term $2x$ from left side to right side, then we will get
\begin{align} & 2x+5y=-16 \\ & \Rightarrow 5y=-2x-16 \\ \end{align}
Dividing the above equation with $5$, then we will get
\begin{align} & \Rightarrow y=\dfrac{-2x-16}{5} \\ & \Rightarrow y=\left( \dfrac{-2}{5} \right)x+\left( \dfrac{-16}{5} \right) \\ \end{align}
Here the slope of the equation $2x+5y=-16$ is $\dfrac{-2}{5}$, intercept of the equation $2x+5y=-16$ is $\dfrac{-16}{5}$.
Considering the equation $6x+y=20$.
Shifting the term $6x$ from left side to right side, then we will get
\begin{align} & 6x+y=20 \\ & \Rightarrow y=-6x+20 \\ \end{align}
Here the slope of the equation $6x+y=20$ is $-6$, intercept of the equation $6x+y=20$ s $20$.
We can observe that both the given equations have different values of slopes, so the given linear system of equations has one solution.

Note: We can also calculate the solution of the given system. We have the graph of the both the given equations as

In the graph we can also observe that point of intersection which will give the solution of the given linear system of equations.