Question

How do you write a system of equations with one solution, a system of equations with no solution and a system of equations with infinitely many solutions $?$

Answer 1

Here in this question, we explain how a system of equations ${a_1}x + {b_1}y + {c_1} = 0$ and ${a_2}x + {b_2}y + {c_2} = 0$. If this equation has one solution, no solution and infinitely many solutions we have a condition by applying that we can say that how many solutions does the equation have.

Complete step-by-step solution:
An equation of the form $ax + by + c = 0$ where $a$, $b$, $c$ $\in \,R$, $a \ne 0$ and $b \ne 0$ is a linear equation in two variables. While considering the system of linear equations, we can find the number of solutions by comparing the coefficients of the equations. Also, we can find whether the system of equations has no solution or infinitely many solutions by graphical method.
Let us consider the pair of linear equations in two variables x and y.
${a_1}x + {b_1}y + {c_1} = 0$
${a_2}x + {b_2}y + {c_2} = 0$
Here ${a_1}$, ${b_1}$, ${c_1}$, ${a_2}$, ${b_2}$, ${c_2}$ are real numbers.
If $\left( {\dfrac{{{a_1}}}{{{a_2}}}} \right) \ne \left( {\dfrac{{{b_1}}}{{{b_2}}}} \right)$ then there will be one solution. This type of equation is called a consistent pair of linear equations.
Consider an example
$2y = 4x + 11$
$y = 3x + 5$
Here these systems of linear equations satisfied the condition $\left( {\dfrac{{{a_1}}}{{{a_2}}}} \right) \ne \left( {\dfrac{{{b_1}}}{{{b_2}}}} \right)$, That being said, there is one solution to this problem.
If $\left( {\dfrac{{{a_1}}}{{{a_2}}}} \right) = \left( {\dfrac{{{b_1}}}{{{b_2}}}} \right) \ne \left( {\dfrac{{{c_1}}}{{{c_2}}}} \right)$ then there will be no solution. This type of equation is called an inconsistent pair of linear equations. If we plot the graph, the lines will be parallel.
Consider an example
$y = 4x + 11$
$y = 4x + 5$
Here these systems of linear equations satisfied the condition $\left( {\dfrac{{{a_1}}}{{{a_2}}}} \right) = \left( {\dfrac{{{b_1}}}{{{b_2}}}} \right) \ne \left( {\dfrac{{{c_1}}}{{{c_2}}}} \right)$, both the lines have same slope 4 which means lines are parallel never intersect. That being said, there is no solution to this problem.
If $\left( {\dfrac{{{a_1}}}{{{a_2}}}} \right) = \left( {\dfrac{{{b_1}}}{{{b_2}}}} \right) = \left( {\dfrac{{{c_1}}}{{{c_2}}}} \right)$ then there will be many solution. This type of equation is called an inconsistent pair of linear equations.
Consider an example
$2y = 4x + 11$
$2y = 4x + 5$
Here these systems of linear equations satisfied the condition $\left( {\dfrac{{{a_1}}}{{{a_2}}}} \right) = \left( {\dfrac{{{b_1}}}{{{b_2}}}} \right) = \left( {\dfrac{{{c_1}}}{{{c_2}}}} \right)$, That being said, there is many solutions to this problem.

Note: While solving the system of equations sometimes we get the value of an unknown variable. By the equation we can plot the graph also, if they are parallel to each other then we will not be able to find the solution for the system of equations. Hence the above example represents that the system of equations has no solutions.