Question

How do you solve the system of equations `x-2y=6` and `2x-4y=10`?

Answer 1

No solution.

Explanation:

To solve any two pair of linear equations in two variables,one important method is substitution method under which value of one variable in terms of another is taken and is substituted in second equation.

Another method is drawing a graph of two lines and their point of intersection gives the solution.

Here first equation is `x-2y=6`, which gives `x=2y+6` and putting this value in second equation `2x-4y=10` gives

`2×(2y+6)-4y=10` or

`4y+12-4y=10` i.e.

`12=10`

But this is contradictory and hence we have no solution. In such a case if we use graphical method and draw graph, we get two parallel lines and hence no solution. One can also say equations are inconsistent.

Note - Some times we may get `0=0` iin substitution method. This denotes that the two equations are essentially same and in graphical method the two lines concide and we have infinite solutions, as we have infinite common points. In case we do not get either we definitely get a clear value of one variable, which when put in another equation gives us single solution. This in graphical method indicates two intersecting lines and point of intersection is the single solution.

Answer 2

The equation cannot be solved and there are no solutions.

Explanation:

`x-2y=6" "` and `" "2x-4y=10 rarr div 2`
`x-2y=6" "` and `" "x-2y=5`

Looking at these equations we should realise that there is a problem!

The left hand side of each equation is the same, but the right hand sides are different. Huh???

We cannot do the same operations with the variables, but get two different results.....

Let's solve anyway.

`x-2y=6" "` and `" "x-2y=5`

It follows that: if `" "x-2y = x-2y" "`
`color(white)(xxxxxxxxxxxxx` then `6=5`

`6 = 5` is clearly a false statement and there are no variables to solve for.

This tells us that the equation cannot be solved and there are no solutions.