How do I solve a linear system

Question

In Algebra 1 we're working on solving linear systems, to problems like "x=y+3 2x-y+5" And I don't even know where to start.

Answer 1

You'll learn better techniques for higher order systems, but the 2nd order system (2 eqtns with 2 unknowns) is straightforward to solve by hand.  Here are the general steps

 

1) Place both equations in the form aX + bY = c

2) Inspect the constants of both equations to find a GCM for either X or Y

3) Multiply Either equation (or both) to achieve the GCM in both eqtns for that X or Y

4) Add or subtract the equations to get RID of that X or Y

5) Solve for that remaining Y or X

6) Substitute that first answer (YorX) to compute the other (XorY)

 

Let's do your example

 

1)

 x-y=3

2x-y=5

 

2)Looking at the above, you'll note that if we subtract the 1st equation from the 2nd, we can make the Y-variable disappear.  To be clear, I'll rewrite them

 

  2x-y=5

-( x-y)=3

x = (5-3)=2

 

6) Now just pick an equation to substitute back to find y (doesn't matter which one you use)

 

-y=3-x    (moved x to the right on 1st eqtn)

 y=x-3    (multiplied by -1 on both sides)

 y=(2)-3= -1  (substituted 2 into x from above)-----> (x,y)=(2,-1)

 

6) Lets check the other eqtn just to verify

 

-y =-2x + 5

y= 2x - 5  (multiplied by -1 on both sides)

y=2(2)-5= -1 (substituted 2 into x from above)-----> (x,y)=(2,-1)  VERIFIED!

 

 

 

Answer 2

Ok as you have said this is linear equation. The first step is to take "x" and "y" to one side for both the equation so you get: x-y = 3 (subtracting y on both side) and 2x-y = -5 ( subtracting 5 both side). Now you subtract the two equation you get : -x = 8, so x = -8 and putting this "x" in first equation gives -8 - y =3. So y= -11. Thank You

Answer 3

Instead of adding equations you could also just use substitution since we know x=y+3.  Plug in x in the 2nd equation and we get

 

2(y+3)-y=5            Solve for y.

2y+6-y=5          

y+6=5

y=-1

 

Now plug in y in the 1st equation and we get

 

x=(-1)+3=2

 

Hope this helps!

Answer 4

X = y +3

2X - y = 5

 

 

  X - y = 3   (1)

 2X - y = 5  ( 2)                       / first make equations in the standard form of

                                         / aX + b =c

 

   You see here that if you subtract equation equ  (1) from (2)

 

  X = 5-3 = 2

 

   Substitute  X =2 back to equation (1)

     

       2 - y = 3       Y = 2 - 3 = -1