Gerard M. answered • 12/08/20
Enthusiastic Math Tutor With 4 Years Experience
These are great questions! You've recognized that there isn't a "one-solution-fits-all" to these problems, so now the next step is to learn when to use what.
1. When is it best to solve a linear system graphically?
Simply put, it's best to solve graphically if you can't (or don't want to) do it algebraically, which is actually most linear systems you can think of! For example, I can think of a random linear system as such:
13x + 37y = 39
21x + 33y = 83
Would you like to solve this by substitution or elimination? Probably not! So let the graphing calculator do it for you. Again, most linear systems are like this. The problems you get that are easily solvable by algebraic methods are specifically chosen because they're easily solved that way!
2. When is it best to solve a linear system by substitution?
When it's easily solvable by substitution! (You might be noticing a theme here) For example:
3x - 7y = 13
y - x = 5
In this example, you could solve by elimination a number of ways, easiest being multiplying the bottom equation by 3, cancelling out x, and then adding them. But that would require reordering the terms, multiplying the entire equation, and keeping track of all of that. Alternatively, you could solve for y or x in the bottom equation, and then substitute it in for the top. This is the method I usually use for this type of problem where x and y have a coefficient of 1, but hey, if you can do elimination quicker and it's easier, go for it! Speaking of elimination...
3. When is it best to solve a linear system using elimination?
Take a look at this system:
5x + 9y = 10
5x + 4y = 5
Isn't it just begging to be eliminated? No need doing all that substitution when a quick subtraction will get rid of that 5x. In general, I tend to like elimination over substitution. When elimination is easy, it's really easy, whereas substitution always has a bit of overhead, like having to distribute after you substitute. Once again, it's all about making things easier for yourself.
4. How can systems of linear equations that have no solution or infinitely many solutions be recognized graphically and algebraically?
What does a "solution" mean for a system of equations? It basically just means where one equation equals the other equation, and that "where" is defined by the x and y values that you plug into the equation. In other words, the solution is an xy-pair that makes both equations the same. Graphically, this solution is the xy-coordinate (or coordinates) where the line representing each equation meet.
So, in what cases would two equations have no solution or infinitely many? Well, graphically, no solution would mean that the lines never touch. First, imagine what that would look like: it looks like two parallel lines! What makes them parallel? This is the algebraic intuition: both lines have the same steepness, or slope, but they "start" from different points. So if we take the slope-intercept form of a line, y = mx + b, the slope m is the same, but they hit the y-intercept ("start") at different points, so b is different. Therefore, a system like
y = 3x + 5
y = 3x + 8
has no solutions because they're parallel, same slope, but they go through different points. How about infinitely many? That means the two lines are actually one and the same! If you graph them, they coincide, they graph the same line. But when you look at equations for lines, be careful, because even though they might look different, they may be the same. Such as:
y = 3x + 5
2y = 6x + 10
Yes, they are different numbers, but if we divide the second equation by 2, we end up with y = 3x + 5 again! This concept is similar to equivalent fractions: the two equations are different representations of the same line. And this applies to the "no solution" case as well! So be sure to identify whether or not the lines are really different, or just different representations.
Hopefully that makes things clearer and satiates your curiosity. Good luck in your studies!
