Question

How do you solve $y = 4x - 3$ and $y = 1$ by graphing and classifying the system?

Answer 1

Hint: In the question, we are provided with two equations that represent straight lines on the Cartesian plane. We plot the graphs of the straight lines by finding the coordinates of some points lying on the line. We substitute the value of $x$ and we determine the value of $y$ and then we mark the points in the graph and we join the points.

Complete step by step answer:
So, we have the equations: $y = 4x - 3$ and $y = 1$.
So, we have to graph the straight lines on the Cartesian plane.
First, we find the coordinates of points lying on the line $y = 4x - 3$.
We find the values of y by using the graph equation $y = 4x - 3$. Let us substitute the value of x as $0$, $1$, and $\dfrac{1}{2}$.
Now we consider the value of x as $0$, the value of y is
\[ \Rightarrow y = 4\left( 0 \right) - 3\]
\[ \Rightarrow y = - 3\]
Now we consider the value of x as $1$, the value of y is
\[ \Rightarrow y = 4\left( 1 \right) - 3\]
\[ \Rightarrow y = 4 - 3\]
\[ \Rightarrow y = 1\]
Now we consider the value of x as $\dfrac{1}{2}$, the value of y is
\[ \Rightarrow y = 4\left( {\dfrac{1}{2}} \right) - 3\]
\[ \Rightarrow y = 2 - 3\]
\[ \Rightarrow y = - 1\]
Now we draw a table for these values we have

x	$0$	$1$	$\dfrac{1}{2}$
y	$ - 3$	$1$	$ - 1$

Similarly, we have to find the coordinates of points lying on the line $y = 1$.
Let us substitute the value of x as $0$, $1$, and $2$.
Now we consider the value of x as $0$, the value of y is
\[ \Rightarrow y = 1\]
Now we consider the value of x as $1$, the value of y is
\[ \Rightarrow y = 1\]
Now we consider the value of x as $2$, the value of y is
\[ \Rightarrow y = 1\]
Now we draw a table for these values we have

x	$0$	$1$	$2$
y	$1$	$1$	$1$

Now, we plot the graphs for both the equations. So, we get,

Now, we can see that the straight lines represented by the two equations intersect each other at a single unique point. So, the system of equations is consistent and possesses a unique solution.

Note: The graph plotted is of two dimensional. The graph is plotted in x-axis versus y axis. By the equation of a graph, we can plot the graph by assuming the value of $x$ and finding the corresponding values of $y$. Consistent system of equations is the system of equations that possess a solution, either a unique solution or infinitely many solutions.