Question

How do you solve the system by graphing: \[y=2x+1\] and \[y=2x-2\]?

Answer 1

Hint: Draw the graph of the two given equations. To draw the graph of a straight line, we need at least two points. So, choose one of the equations and substitute x = 0, determine y, then substitute y = 0, determine x. Now, apply the same process for the second equation. Plot the graph of the two equations using the points obtained. Check the point of intersection to get the answer. If there are no intersection points then there will not be any solution.

Complete step-by-step solution:
Here, we have been provided with the system of equations: \[y=2x+1\] and \[y=2x-2\] and we are asked to solve it using the graph.
Now, let us assume the two equations as: -
\[\Rightarrow y=2x+1\] - (1)
\[\Rightarrow y=2x-2\] - (2)
Let us consider equation (1), so we have,
\[\Rightarrow y=2x+1\]
Substituting x = 0, we get,
\[\Rightarrow y=1\]
Substituting y = 0, we get,
\[\begin{align}
  & \Rightarrow 2x+1=0 \\
 & \Rightarrow x=\dfrac{-1}{2} \\
\end{align}\]
Therefore, the points where the line will cut the axes are: - A (0, 1) and \[B\left( \dfrac{-1}{2},0 \right)\].
Let us consider equation (2), we have,
\[\Rightarrow y=2x-2\]
Substituting x = 0, we get,
\[\Rightarrow y=-2\]
Substituting y = 0, we get,
\[\begin{align}
  & \Rightarrow 2x-2=0 \\
 & \Rightarrow 2x=2 \\
\end{align}\]
\[\Rightarrow x=1\]
Therefore, the points where the line will cut the axes are: - C (0, -2) and D (1, 0).
So, the graph of the two linear equations can be plotted as: -

From the above graph we can clearly see that the two straight lines are parallel to each other and they are not intersecting at any point. So, we can conclude that the given system of linear equations has no solution.

Note: One may note that we can check our answer by solving the equations of the two given lines algebraically using the substitution method. If we will get the same coordinate of P as in the graph then our answer will be correct. Remember that while drawing the graph, substitute x = 0 and y = 0 to determine the points or you may have to perform some calculations while determining the points. You must remember the conditions for a system to have a solution. Remember that if two lines are parallel then they will have no solutions, if they will intersect at a particular point then there will be a unique solution and if they will overlap then they will have infinite number of solutions.