Question 167828: Solve each system by the substitution method.
x – y = 5
2x = 2y + 14
3x + y = 2
-x – 3y = 6
3x – 4y = 9
-3x + 4y = 12
Answer by midwood_trail(310)   (Show Source):
You can put this solution on YOUR website! This is not linear algebra. This is a topic given in algebra 2 called solving a system of linear equations in two variables. You will not see linear algebra unless you decide to major in math. Linear algebra is often given after the entire calculus series.
I will solve one of the questions.
Solve each system by the substitution method.
x – y = 5
2x = 2y + 14
To substitute means exactly that-->replace one letter for another and simplify to find the other.
I will give each equation a name to help me follow the steps.
I will call this equation x – y = 5...Batman.
I will call the other equation 2x = 2y + 14...Robin.
I will solve for x in equation Batman as my first step.
x - y = 5
x = y + 5
Do you see why I did?
I found that x = y + 5 and so now I will go into equation Robin and substitute x with y + 5 to find the value of y (if it can be found).
Here is equation Robin: 2x = 2y + 14
Let x = y + 5
2(y + 5) = 2y + 14
2y + 10 = 2y + 14
We now solve for y. Once we find y, we are very close to the end.
We now combine like terms.
2y - 2y = -10 + 14
Here is an interesting thing.
On the left side of the equation, we learn that 2y and -2y become zero.
What does that mean?
It means that there is NOTHING more that we can do with this first system of equations.
In other words, the answer to the first system of equations in two variables is NO SOLUTION or empty set.
You will come across many equations that have no solutions.
Math teachers love to throw this type of question on exams as curves to trick students.
Since there is no solution for y, we cannot find x and vice-versa.
Final answer: NO SOLUTION or EMPTY SET.
Do you understand what I did?
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