Q4. - Algebra 1

Question

Graph each system and determine a number of solutions that it has. If it has one solution, name it.

$\begin{array}{c}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{4}\\ \mathbf{2}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{1}\end{array}$

Step-by-Step Solution

Answer

The graph of the given system of equations is:

The number of solutions that the given system of equations has is zero that implies the given system of equations has no solution.

Step 1. Graph the given system of equations.

The given system of equations is:

$\begin{array}{c}2x+3y=4\\ 2x+3y=-1\end{array}$

The equation is: $2x+3y=4$

Substitute 0 for $x$ and find the value of $y$.

$\begin{array}{c}2x+3y=4\\ 2\left(0\right)+3y=4\\ 0+3y=4\\ 3y=4\\ y=\frac{4}{3}\end{array}$

Therefore, one of the point is $\left(0,\frac{{\displaystyle 4}}{{\displaystyle 3}}\right)$

Substitute 0 for $y$ and find the value of $x$.

$\begin{array}{c}2x+3y=4\\ 2x+3\left(0\right)=4\\ 2x=4\\ x=\frac{4}{2}\\ x=2\end{array}$

Therefore, the other point is $(2,0)$

Therefore, the graph of the equation $2x+3y=4$ is the line passing through the points $\left(0,\frac{{\displaystyle 4}}{{\displaystyle 3}}\right)$ and $(2,0)$.

The equation is: $2x+3y=-1$

Substitute 0 for $x$ and find the value of $y$.

$\begin{array}{c}2x+3y=-1\\ 2\left(0\right)+3y=-1\\ 3y=-1\\ y=\frac{-1}{3}\end{array}$

Therefore, one of the point is $\left(0,\frac{{\displaystyle -1}}{{\displaystyle 3}}\right)$

Substitute 0 for $y$ and find the value of $x$.

$\begin{array}{c}2x+3y=-1\\ 2x+3\left(0\right)=-1\\ 2x=-1\\ x=\frac{-1}{2}\end{array}$

Therefore, the other point is $\left(\frac{{\displaystyle -1}}{{\displaystyle 2}},0\right)$

Therefore, the graph of the equation $2x+3y=-1$ is the line passing through the points $\left(0,\frac{-1}{3}\right)$ and $\left(\frac{{\displaystyle -1}}{{\displaystyle 2}},0\right)$.

 

Therefore, the graph of the given system of equations is:

Step 2. Determine the number of solutions that the given system of equations has.

From the graph of the system of equations, it can be noticed that there is no intersection point of the lines as the lines are parallel lines. Therefore, the given system of equations has no solution.