SOLUTION: Helping my son with some algebra stuff. We're struggling with systems of equations. What we're struggling with mostly is 'why' you do the order of operations in the way they are do

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Question 1116936: Helping my son with some algebra stuff. We're struggling with systems of equations. What we're struggling with mostly is 'why' you do the order of operations in the way they are done. Here's the problem:
a-6=2b
a+b=15
The questions calls for the point where the these equations cross (#,#)
Here's where we are:
Consolidating the problems: a+2b+a+b=6+15 or 2a+3b=21
Using the basic y=Mx+b to get slope.
Now have no clue where to go from here.
Appreciate the help
Robert & Jason
Found 2 solutions by solver91311, ikleyn:
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!


The first step you took was an error, so go back to the beginning. Since your equations are in "a" and "b", they represent lines on an "ab" coordinate plane rather than an "xy" coordinate plane. The choice of which variable is on the horizontal axis and which is on the vertical is completely arbitrary and won't change the final outcome, so just for the sake of neatness, let's assume that ordered pairs in this plane will be represented in alphabetical order, that is to say a point in the plane, rather than being specified as will be specified as

Given that, the first step is to specify each of your equations as b being a function of a, in other words, put them in the form , where is the slope and is the b-axis intercept.





Then





Since these two equations have different slope values, they must perforce intersect somewhere, and this is at the point where the values of a and b satisfy both equations.

Since we have two expressions in "a" that are equal to the variable b, set these two expressions equal to each other and solve for a.











Hence, the point of intersection is the ordered pair

John

My calculator said it, I believe it, that settles it
Answer by ikleyn(52818)   (Show Source): You can put this solution on YOUR website!
.
a - 6 = 2b     (1)
a + b = 15     (2)


    "The questions calls for the point where these equations cross (#,#)".


The question CAN NOT be "for the point where these equations cross (#,#)".  // Since equations do not cross.


The question can be EITHER to find the intersection point of the two lines (1) and (2) OR to find the common solution to this system of equations.


Both these questions are equivalent.


To solve the system,  express  "a" from equation (1):  a = 2b + 6, and then substitute it into equation (2) to get

(2b+6) + b = 15

3b + 6 = 15

3b = 15-6  ====>  3b = 9  ====>  b =  = 3.


Now, when you just found the value of b = 3, substitute it into equation (1) to find  a = 2b+6 = 2*3 + 6 = 12.


Answer.  The solution to the system  is a= 12,  b= 3,  or the pair (a,b) = (12,3).


Check.   Check the solution/(the answer) on your own by substituting the found values into original equations.


The method I used in the solution  is called  "the Substitution method".

It is  THE FIRST method  that students use when they start learning solving systems of equations.



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