Question

Solve the following system of linear equations graphically. \[3x + 2y - 4 = 0\] and \[2x - 3y - 7 = 0\]. Shade the region bounded by the lines and the x-axis.

Answer 1

Hint: Solve the two equations by putting \[x = 0\] and find value for \[y\] and then putting \[y = 0\] and finding value of \[x\]. You will get two points for each equation respectively. Draw the tables showing the respective points for both the equations separately and then mark them on the graph.

Complete step-by-step solution:
We are given the equations:
\[3x + 2y - 4 = 0\] … (1)
\[2x - 3y - 7 = 0\] … (2)
Now putting \[x = 0\] in equation (1) we get
\[3 \times 0 + 2y - 4 = 0\]
Hence we get
\[2y - 4 = 0\]
Taking the constant term to another side we get
\[2y = 4\]
Dividing both sides by \[2\] we get
\[y = 2\]
Therefore we get \[x = 0,y = 2\]
Now putting \[y = 0\] in equation (1) we get
\[3x + 2 \times 0 - 4 = 0\]
Hence we get
\[3x - 4 = 0\]
Taking the constant term to another side we get
\[3x = 4\]
Dividing both sides by \[3\] we get
\[x = \dfrac{4}{3}\]
Therefore we get \[x = \dfrac{4}{3},y = 0\]
Use the following table to draw the graph:

x	\[0\]	\[\dfrac{4}{3}\]
y	\[2\]	\[0\]

The graph of (1) can be obtained by plotting the two points \[A(0,2),B\left( {\dfrac{4}{3},0} \right)\]
Now putting \[x = 0\]in equation (2) we get
\[2 \times 0 - 3y - 7 = 0\]
Hence we get
\[ - 3y - 7 = 0\]
Taking the constant term to another side we get
\[ - 3y = 7\]
Dividing both sides by \[ - 3\] we get
\[y = - \dfrac{7}{3}\]
Therefore we get \[x = 0,y = - \dfrac{7}{3}\]
Now putting \[y = 0\] in equation (2) we get
\[2x - 3 \times 0 - 7 = 0\]
Hence we get
\[2x - 7 = 0\]
Taking the constant term to another side we get
\[2x = 7\]
Dividing both sides by \[2\] we get
\[x = \dfrac{7}{2}\]
Therefore we get \[x = \dfrac{7}{2},y = 0\]
Use the following table to draw the graph:

X	\[0\]	\[\dfrac{7}{2}\]
y	\[ - \dfrac{7}{3}\]	\[0\]

The graph of (2) can be obtained by plotting the two points \[C\left( {0, - \dfrac{7}{3}} \right),D\left( {\dfrac{7}{2},0} \right)\]

Note: A single linear equation in two variables may have infinite solutions. A pair of linear equations can have no solution, a unique solution and infinitely many solutions depending upon the conditions of the coefficients of the variables. The general form of a linear equation in two variables is $ax + by + c = 0$. If \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}\], then the two equations have infinitely many solutions. If \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}\], then the two lines have no solution at all. If \[\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}\], the two equations have a unique solution.