Question

One equation in a system of linear equation is y=-2x+4.What is a second equation for the system so that the system only has one solution?

Answer 1

The second equation for the system so that the system only has one solution is
`y = (1)/(2) (x - k)` or x – 2y = k, where "k" is any constant

Solution:

Given that, One equation in a system of linear equation is y = -2x + 4

`\rightarrow 2 x+y=4 \rightarrow(1)`

We have to find what is a second equation for the system so that the system only has one solution?

Now, we know that, perpendicular lines will have only one solution, so let us find the perpendicular line to the given equation.

First let us find the slope of given line,

`\text { Slope } m=\frac{-x \text { coefficient }}{y \text { coefficient }}=(-2)/(1)=-2`

And, product of slopes of perpendicular lines = - 1

`\begin{array}{l}{\rightarrow-2 * \text { slope of perpendicular line }=-1} \\\\ {\rightarrow \text { slope of perpendicular line }=(1)/(2)}\end{array}`

Now, line equation in slope intercept form is y = mx + c

where "m" is the slope of line

`\begin{array}{l}{\rightarrow y=(1)/(2) x+c} \\\\ {\rightarrow 2 y=x+2 c} \\\\ {\rightarrow x-2 y=-2 c}\end{array}`

Let k = -2c, hence the above equation becomes,

x – 2y = k

`y = (1)/(2) (x - k)`

"k" can be any constant

The second equation for the system so that system only has one solution is found out