Question

How do you write a system of equations with the solution \[\left( {4, - 3} \right)\]?

Answer 1

Hint: Here, we will substitute different values in the coefficients of \[x\] and \[y\] to find two equations out of infinitely many equations with this particular solution. Substituting \[\left( {x,y} \right)\] as \[\left( {4, - 3} \right)\], we will be able to find the constant of these equations and by substituting them, we will be able to find the required system of equations.

Complete step-by-step answers:
In order to write a system of equations with the solution \[\left( {4, - 3} \right)\], we should know that there are infinitely many equations having this particular solution.
Thus, as we know that the standard form of a linear equation in two variables is \[ax + by + c = 0\] or \[ax + by = - c\], we will choose any coefficients for \[x\] and \[y\] and try to make two equations keeping in mind that the coefficients are not proportional.
Thus, let \[a = 1\] and \[b = 2\]
Hence, we get the first equation as:
\[1x + 2y = c\]
And for the second equation let the coefficients \[a,b\] be equal to \[3,4\] respectively.
Thus, we get,
\[3x + 4y = k\]
Now, since, the solution of these equations should be \[\left( {4, - 3} \right)\]
Hence, substituting \[x = 4\] and \[y = - 3\] in \[1x + 2y = c\], we get,
\[\left( 1 \right)\left( 4 \right) + 2\left( { - 3} \right) = c\]
Multiplying the terms, we get
\[\begin{array}{l} \Rightarrow 4 - 6 = c\\ \Rightarrow - 2 = c\end{array}\]
Hence, the first equation is:
\[x + 2y = - 2\]
Now, substituting \[x = 4\] and \[y = - 3\] in \[3x + 4y = k\], we get,
\[3\left( 4 \right) + 4\left( { - 3} \right) = k\]
Multiplying the terms, we get
\[\begin{array}{l} \Rightarrow 12 - 12 = k\\ \Rightarrow 0 = k\end{array}\]
Hence, the first equation is:
\[3x + 4y = 0\]
Therefore, these are two systems of equations which are having the solution \[\left( {4, - 3} \right)\]
By assuming various values of the coefficients of \[x\] and \[y\], we can find infinitely many systems of equations having this solution.

Note:
An equation is called linear equation in two variables if it can be written in the form of \[ax + by + c = 0\] where \[a,b,c\] are real numbers and \[a \ne 0\] , \[b \ne 0\] as they are coefficients of \[x\] and \[y\] respectively. Also, the power of linear equations in two variables will be 1 as it is a ‘linear equation’. Also, a linear equation in two variables can sometimes have infinitely many solutions rather than only one in the case of ‘one variable’.