Name: Cross Multiplication Method: Illustration
Uploaded: 2019-07-19
Duration: 7 min 27 s
Description: This tutorial will show you many examples of solving linear equations in two variables using the cross-multiplication method with a clear step-by-step explan...

Question

For what value of $k$ does the system of equations
$x + 2 y = 3, 5 x + k y + 7 = 0$
have no solution?
Also, show that there is no value of $k$ for which the given system of equation has infinitely many solutions.

Accepted Answer

The given system of equations are,
$x + 2 y = 3$

$x + 2 y - 3 = 0 \dots . (i)$
And, $5 x + k y + 7 = 0 \dots (i i)$
These equations are of the following form
$a_{1} x + b_{1} y + c_{1} = 0$ , $a_{2} x + b_{2} y + c_{2} = 0$
where, $a_{1} = 1, b_{1} = 2, c_{1} = - 3$ and $a_{2} = 5, b_{2} = k, c_{2} = 7$
In order that the given system of equations has no solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
$\frac{1}{5} = \frac{2}{k} \neq \frac{- 3}{7}$
$k = 10$ , $k \neq \frac{- 14}{3}$
Hence, the required value of $k$ is $10$ for

Condition for having infinite many solution
$a_{1} = 1, b_{1} = 2, c_{1} = - 3$ and $a_{2} = 5, b_{2} = k, c_{2} = 7$
If we put $k = 10$ , then $\frac{a_{1}}{a_{2}}$ will become equal to $\frac{b_{1}}{b_{2}}$ , but there is no such value to equalise $\frac{b_{1}}{b_{2}}$ and $\frac{c_{1}}{c_{2}}$
So, for given case, $\frac{1}{5} = \frac{2}{10} \neq \frac{- 3}{7}$
There is no value of $k$ for which the given system of equations has an infinite number of solutions.

For what value of $k$ does the system of equations
$x + 2 y = 3, 5 x + k y + 7 = 0$
have no solution?
Also, show that there is no value of $k$ for which the given system of equation has infinitely many solutions.

Important Questions on Linear Equations in Two Variables

Learn and Prepare for any exam you want !

For what value of k does the system of equations x + 2y = 3, 5x + ky + 7 = 0 have no solution? Also, show that there is no value of k for which the given system of equation has infinitely many solutions.

Important Questions on Linear Equations in Two Variables

Learn and Prepare for any exam you want !

For what value of $k$ does the system of equations
$x + 2 y = 3, 5 x + k y + 7 = 0$
have no solution?
Also, show that there is no value of $k$ for which the given system of equation has infinitely many solutions.