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# Solving systems of linear equations — Basic example

Watch Sal work through a basic Solving systems of linear equations problem.

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• But how does this work because if you plug in the x(3y) into the top equation you get y=9y.
• Actually, these two lines intersect at the origin (as Sal mentioned). I originally got y = 9y and thought that there were no solutions, but if you subtract y from both sides you get 0 = 8y. Divide both sides by 8 and you'll still get 0 for y. The mistake that we made was by not combining like terms.
• : IQ increasing to -14:
• actually, the lessons are very understandable
• So to summarize:
if mx and b is exactly the same = infinitely many solutions
if mx is reciprocal = one solution
if mx is the same = no solution
• Actually, If the two equations are similar for slope and y-intercept, then it has infinitely many solutions. If the slope is different for the two equations with similar y-intercepts, then it is one solution. If the slope of the two is the same but different y-intercepts, then it is no solution.
• can we solve these equations to get the ans?
• Yes, but you have to be careful. The answer is not the number that you would get when you solved the equations. The answer is `whether` there are two solutions, only one solution, no solutions or infinite solutions.
If you are able to get any solution, you CAN say that the "zero solutions option" is not correct
Another reason it would be tricky to solve is that this problem is tricky to solve. If you try substitution to solve this system, you get some strange equations.
x = 3y
y = 3x

If you substitute the value for x into the equation for y you get y = 3 (3y)
So y = 9y

You might be tempted to say IMPOSSIBLE! y cannot equal 9y
Instead you need to use algebra to isolate the y by subtracting y from both sides:
y - y = 9y - y
So 0 = 8y
divide both sides by 8 to get `y = 0`
plug that into your original equation to find out that when y = 0, x = 0
So there is one solution and it also explains why y can equal 9y.

Sal decided to use the fact that this is a system of linear equations, which means it represents two lines. He quickly graphed the lines to show that they intersect at (0, 0) which is the same solution we found be solving.
Since they are two lines, they CANNOT have two solutions--two lines can cross one time or can coincide (be duplicates of the same line) in which they have the same solutions at an infinity of places. If the lines are parallel (never crossing) they have NO solutions.

Graphing is probably the easiest, safest and fastest way to answer this question, but solving gave us the same answer.
• if both equations have the same slope, how would i know whether they are parallel or the same line....?
• Lines are defined based on their slope and y-intercept. If two lines have the same slope, the thing that differentiates them is their y-intercept. Lines that are the same will have the same slope and y-intercept, and lines that are only parallel will have the same slope but different y-intercepts.
For example, lines (1) and (2) below are the same, but (3) is just parallel to them and not the same.
(1) y = 2x + 4
(2) 2y - 8 = 4x
(3) y = 2x - 2
• So if both equations have the same slope, how would i know whether they are parallel or the same line??
• The slope formula is y=mx+b.
m is the slope and b is the y axis intercept.

To determine weither they are parallel or the same line we look at the y axis intercept (b).
If they have the same value then they are the same line and if they're different then they are parallel giving that both equations have the same slope.
• To sum on how to know know how many solutions a line has -

*For context: Slopes are defined as MX and y-intercepts as B in the slope-intercept equation (y = mx + b)

1. If the slopes are different, it has one solution

2. If the slopes and y-intercept is the same, infinite solutions

3. If the slopes are the same but different y-intercepts, it has no solutions

**If confused, use the desmos calculator by plugging in values and checking how the lines look :D

Cheers! ATB & Do ask if you got any Q's - will try my best to answer it <3