Question

For `2x+4y=10` and `2x+4y=-10`, there is/are 1. 1 solution, 2. 2 solutions, 3. infinitely many solutions, 4. no solutions?

Answer 1

No solution - the lines are parallel

Explanation:

Let's graph the two of them and see what happens:

graph{(2x+4y-10)(2x+4y+10)=0 [-18.02, 18.02, -9.01, 9.01]}

They are parallel lines and so it's option 4 - no solution.

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Let's look at these equations a different way. I'm going to change them into slope-intercept form, where the general formula is:

`y=mx+b; m="slope", b=y"-intercept"`

`2x+4y=10=>y=-1/2+5/2`

`2x+4y=-10=>y=-1/2-5/2`

And so we can see that the slope of these two lines is the same by the point where they intersect the `y` axis is different, just as shown in the graph.

Answer 2

4) There is no solution.

Explanation:

Just by looking at the given equations you should be able to see that there is a problem with the equations:

Both left sides of the equations are equal, but the right sides are not:

`color(red)(2x +4y) = color(lime)(10)`
`color(red)(2x +4y)= color(magenta)(-10)`

It is not possible to add 2 identical terms and get different answers.

Mathematically:
If`" "2x +4y = 2x+4y`

Then`" " 10= -10" "larr` this is clearly false

This indicates that there is no solution for the system of equations.