What makes an equation equivalent to another?

[Atran]

I know that this is basic, but I don't get it. Say we have the system,
$x + y = 32$
$3x + 2y = 70$
Each equation represents a graph, and the solution is the point where the two lines intersect.

Here is where I am confused:
$x + y = 32$
$2x + 2y = 64$
What is the proof that the two equations are equivalent / represent the same graph?

I read that two equations are equivalent if they share the same implication:

Equivalent equations are ones such that the truth of one implies and is implied by the truth of the other.

Thanks for help.

 

[jedishrfu]

Start by saying that the two equations represent two distinct lines.

What can you conclude from that?

 

[Mark44]

I know that this is basic, but I don't get it. Say we have the system,
$x + y = 32$
$3x + 2y = 70$
Each equation represents a graph, and the solution is the point where the two lines intersect.

Here is where I am confused:
$x + y = 32$
$2x + 2y = 64$
What is the proof that the two equations are equivalent / represent the same graph?

I read that two equations are equivalent if they share the same implication:

Equivalent equations are ones such that the truth of one implies and is implied by the truth of the other.

Thanks for help.

Equivalent equations have the same solution set.

Your first two equations are not equivalent, since their graphs are different. They do intersect at a point, though. Assuming that they intersect at (2, 3), the equations x = 2 and y = 3 would be equivalent to the first pair of equations you have above. (Note that I just made up a solution, so my equations x = 2 and y = 3 aren't actually equivalent to yours.)

For your second system of equations, both equations represent exactly the same line. Each solution of the first equation is also a solution of the second equation. In different words, if (x₀, y₀) is a solution (i.e., makes the equation a true statement) of the first equation, it will also be a solution of the second equation. For this example, the two equations are equivalent.

 

[Atran]

So there are two ways of thinking:

1) If the two equations have one solution (which is one pair (x_i, y_i)) then I treat x and y in the equations as two (different) values.
$x+y=32$
$3x+2y=70$

$2x+2y=64$
$3x+2y=70$

$(3x+2y) - (2x+2y) = 70 - 64$
$x = 6$ and $y = 26$

2) (x+y=32) is equivalent to (2x+2y=64), since they share the same set of coordinates, therefore the same graph. Let's say set A has the coordinates of (x+y=32), and set B of (3x+2y=70). The solution is the set, $A \cap B$.

 

[HallsofIvy]

Two equations are equivalent if they have exactly the same solution set. If (x, y) is in the solution set of the first equation then x+ y= 32. Multiplying both sides of the equation by 2, 2x+ 2y= 64 so (x, y) is in the solution set for that equation. That proves that the solution set for x+ y= 32 is a subset of the solution set for 2x+ 2y= 64. Now we have to do it the other way: if (x, y) is in the solution set for 2x+ 2y= 64, then they satisfy that equation. Dividing both sides by 2, x+ y= 32 so (x, y) is also in the solution set for that equaton and the solution set for 2x+ 2y= 64 is a subset of the solution set for x+ y= 32. Since each is a subset of the other, they are equal and the two equations are equivalent.