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

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"

$2 x + 4 y = 10 \implies y = - \frac{1}{2} + \frac{5}{2}$

$2 x + 4 y = - 10 \implies y = - \frac{1}{2} - \frac{5}{2}$

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.

Mar 24, 2017

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:

$\textcolor{red}{2 x + 4 y} = \textcolor{\lim e}{10}$
$\textcolor{red}{2 x + 4 y} = \textcolor{m a \ge n t a}{- 10}$

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

Mathematically:
If$\text{ } 2 x + 4 y = 2 x + 4 y$

Then$\text{ " 10= -10" } \leftarrow$ this is clearly false

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