Question
Show graphically that the following systems of equations is in-consistent (i.e. has no solution):
$3\text{x}-4\text{y}-1=0$
$2\text{x}-\frac{8}{3}\text{y}+5=0$
$3\text{x}-4\text{y}-1=0$
$2\text{x}-\frac{8}{3}\text{y}+5=0$
✓
Answer
We have,
$3\text{x}-4\text{y}-1=0$
$2\text{x}-\frac{8}{3}\text{y}+5=0$
Now,
$3\text{x}-4\text{y}-1=0$
$\Rightarrow3\text{x}=1+4\text{y}$
$\Rightarrow\text{x}=\frac{1+4\text{y}}{3}$
When y = 2, we have
$\text{x}=\frac{1+4\times2}{3}=3$
When y = -1, we have
$\text{x}=\frac{1+4\times(-1)}{3}=-1$
Thus, we have the following table giving points on the line 3x - 4y - 1 = 0.
$2\text{x}-\frac{8}{3}\text{y}+5=0$
$\Rightarrow\frac{6\text{x}-8\text{y}+15}{3}=0$
$\Rightarrow6\text{x}-8\text{y}+15=0$
$\Rightarrow6\text{x}=8\text{y}-15$
$\Rightarrow\text{x}=\frac{8\text{y}-15}{6}$
When y = 0, we have,
$\text{x}=\frac{8\times0-15}{6}=-2.5$
When y = 3, we have,
$\text{x}=\frac{8\times3-15}{6}=1.5$
Thus, we have the following table giving points on the line $2\text{x}-\frac{8}{3}\text{y}+5=0.$
Graph of the given equations.
We find the lines represented by equations 3x - 4y - 1 = 0 and $2\text{x}-\frac{8}{3}\text{y}+5=0$ are parallel. So, the two lines have no common point.
Hence, the given system of equations is in-consistent.
$3\text{x}-4\text{y}-1=0$
$2\text{x}-\frac{8}{3}\text{y}+5=0$
Now,
$3\text{x}-4\text{y}-1=0$
$\Rightarrow3\text{x}=1+4\text{y}$
$\Rightarrow\text{x}=\frac{1+4\text{y}}{3}$
When y = 2, we have
$\text{x}=\frac{1+4\times2}{3}=3$
When y = -1, we have
$\text{x}=\frac{1+4\times(-1)}{3}=-1$
Thus, we have the following table giving points on the line 3x - 4y - 1 = 0.
|
x
|
-1
|
3
|
|
y
|
-1
|
2
|
$\Rightarrow\frac{6\text{x}-8\text{y}+15}{3}=0$
$\Rightarrow6\text{x}-8\text{y}+15=0$
$\Rightarrow6\text{x}=8\text{y}-15$
$\Rightarrow\text{x}=\frac{8\text{y}-15}{6}$
When y = 0, we have,
$\text{x}=\frac{8\times0-15}{6}=-2.5$
When y = 3, we have,
$\text{x}=\frac{8\times3-15}{6}=1.5$
Thus, we have the following table giving points on the line $2\text{x}-\frac{8}{3}\text{y}+5=0.$
|
x
|
-2.5
|
1.5
|
|
y
|
0
|
3
|
We find the lines represented by equations 3x - 4y - 1 = 0 and $2\text{x}-\frac{8}{3}\text{y}+5=0$ are parallel. So, the two lines have no common point.
Hence, the given system of equations is in-consistent.
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