In order to find solutions to both equations: 3y-15x = -12 and y-5x = -6,
We need to decide amongst various methods of solving 2 equations of 2 variables. We could use substitution, in which we plug one equation into the other, or we could add the equations, called elimination because it works by eliminating one variable.
I'll show you both methods:
1) Substitution--> since we already have a y alone (no coefficient) in the 2nd equation, we can make that equation "y =", in terms of x. So,
y-5x = -6 --> add 5x to both sides
y-5x +5x = -6 +5x --> y = 5x-6. Now, since y is equal to "5x-6", then we can substitute "5x-6" in anywhere there's a y in the 1st equation.
3y-15x = -12 --> 3(5x-6)-15x = -12
We must take caution to distribute correctly in the next step:
[3×5x + 3×-6] -15x = -12
15x + -18 -15x = -12 --> since 15x and -15x cancel each other out (sum is 0), then that gives us -18 = -12 which is "NO SOLUTION" using that method. One reason two linear equations may have no solution, is if their slopes are equal, meaning that their lines on a graph are parallel. And parallel lines never intersect, whereas solutions of lines, by definition, are where they DO INTERSECT.
2) Now let's try elimination; our goal is to eliminate one variable out of both equations by addition. Start by putting one on topof the other:
3y-15x = -12
y -5x = -6 --> notice that if we were to multiply the bottom equation by (-3), then that would make it:
-3y+15x = 18 --> this also gives us no solution:
3y-15x = -12
-3y+15x = 18
Uh oh... the y's and x's cancel out again, so NO SOLUTION again.
Let's find the slope (m) in slope-intercept graphical form, in which we set each as "y =", and the coefficient of the x term is the slope (m) if the line.
3y-15x = -12--> 3y-15x +15x = -12 +15x
--> 3y = 15x -12--> 3y/3 = (15x-12)/3
--> y = 5x -4 --> this slope is 5
y-5x = -6--> y-5x +5x = -6 +5x
--> y = 5x -6 --> this slope is also 5
Therefore there indeed is NO SOLUTION for these two equations!!
