Why can I solve equations by representing them graphically.

[Cambridge101]

Hi,

Can someone please explain the proof for why 3x+2=8 is the same as saying where the line y=3x+2 is equal to the line y=8.

Cheers.

 

[Dr.Peterson]

Hi,

Can someone please explain the proof for why 3x+2=8 is the same as saying where the line y=3x+2 is equal to the line y=8.

Cheers.

Can you explain where you are confused?

At that intersection point, y = 3x + 2 and also y = 8, so 3x + 2 = 8. The value of x at that point is the solution.

 

[Cambridge101]

Can you explain where you are confused?

At that intersection point, y = 3x + 2 and also y = 8, so 3x + 2 = 8. The value of x at that point is the solution.

Well. When you first use equations, you learn them like this. I had some number x, I multiplied by 3 and added two, this makes 8. Like a puzzle.
3x+2=8, therefore you solve to find x by doing the inverse of what was done to x.

But how does expressing each side of the puzzle as a function solve this. When I then graph 3x+2, I now have an infinite amount of possible co-ordinates that are on this line. So why does making this line, equal to a line y=8, which also has an infinite amount of co-ordinates, do the same thing as the doing the inverse of the puzzle above.

Or, is it simply because we have modelled the puzzle as a function. Where y=8 and y=3x+2 they both = y, share the same y, thus must be equal to each other.

 

[lev888]

Well. When you first use equations, you learn them like this. I had some number x, I multiplied by 3 and added to, this makes 8. Like a puzzle.
3x+2=8, therefore you solve to find x by doing the inverse of what was done to x.

But how does expressing each side of the puzzle as a function solve this. When I then graph 3x+2, I now have an infinite amount of possible co-ordinates that are on this line. So why does making this line, equal to a line y=8, which also has an infinite amount of co-ordinates, do the same thing as the doing the inverse of the puzzle above.

Or, is it simply because we have modelled the puzzle as a function. Where y=8 and y=3x+2 they both = y, share the same y, thus must be equal to each other.

You don't make one line equal to the other. The lines are different. You equate the 2 expressions to find the x that makes the values at that point equal. In terms of graphs, you find the point where the lines intersect.

 

[blamocur]

But how does expressing each side of the puzzle as a function solve this.

Personally, I think it illustrates it rather than solves.
Both lines have infinite numbers of points, but their intersection has only one point, with the X-coordinate giving the solution.
Would it make more sense to you if an equation contained [imath]x[/imath] on both sides? E.g.: [imath]3x+2 = x+6[/imath].

 

[Dr.Peterson]

Well. When you first use equations, you learn them like this. I had some number x, I multiplied by 3 and added two, this makes 8. Like a puzzle.
3x+2=8, therefore you solve to find x by doing the inverse of what was done to x.

That is a process you can use to solve an equation. What a graph does is to make visual what it means for a value of x to be the solution: a value of x for which the two expressions (on each side of the equal sign) are equal. The intersection of the two lines is a point where the two values of y (that is, the values of the two expressions) are equal for the same value of x.

It's important to realize that there can be many ways to find a solution, though there is only one solution (for this type of equation). The solution is not the same as the solving process.

But how does expressing each side of the puzzle as a function solve this. When I then graph 3x+2, I now have an infinite amount of possible co-ordinates that are on this line. So why does making this line, equal to a line y=8, which also has an infinite amount of co-ordinates, do the same thing as the doing the inverse of the puzzle above.

Or, is it simply because we have modelled the puzzle as a function. Where y=8 and y=3x+2 they both = y, share the same y, thus must be equal to each other.

Yes, you could say it that way, if you think of an equation as a puzzle. (Actually it's two functions, though in this case one is very simple!)

But the graph doesn't "do" the same thing as the algebraic solving process; if anything, it provides an alternative method of solution, allowing you to look along the one line until you find a place where its y is equal to the other. Once you've found it, you really need to check that the value of x that you found actually satisfies the equation, especially if the graph is rough or the coordinates are not easy to read.