solve each system by substitution

Question

I neeed help on how to solve each system by substitution

 

 

(y=-6                        (-x=y=-4                     (x-3y=3

(y=-2x-2                   (y=2x-11                     (2x=3y    

Answer 1

Hello Eric,

[  y = - 6     and    y = -2x - 2   ]

In this problem, the first equation is solved from the very beginning so we have no choice but to start with it.

y = - 6   <<<  found y !

That was a little too easy, huh?  Now take -6 and substitute it for y in the other equation...

 y = -2x - 2
-6 = -2x - 2
-6 + 2 = -2x
    -4   = -2x
       2 = x       <<< found x !

So our solution to the system is

[ x = 2 ; y = - 6]
—————————



Next problem I think you made a typo. Compare my starting equations with the printed problem.

[ -x + y = 4 and y = 2x - 11 ]

Here we can choose which equation to start with and which variable to solve for first. Let’s start with the first equation and solve for y first. Why? Why not!?!

-x + y = 4
y = 4 + x <<< found expression for y.

Now substitute our expression for y into the OTHER equation.

y = 2x - 11
4 + x = 2x - 11
x - 2x = -11 - 4
-x = -15
x = 15 <<< found x !

Now substitute the answer for x, into EITHER of the equations. x = 15

-x + y = 4 y = 2x - 11
-(15) + y = 4 y = 2(15) - 11
y = 4 + 15 y = 30 - 11
y = 19 y = 19

found y !

So the solution to our system is

[ x = 15 ; y = 19 ]
——————————


Last one…

[ x - 3y = 3 and 2x = 3y ]

I pick the first equation solved first for y. Solving the second equation first leads to fractions; yuck!

x - 3y = 3
x = 3 + 3y <<< found expression for x.

Substitute expression for x into the other equation…

2x = 3y
2(3 + 3y) = 3y
6 + 6y = 3y
6y - 3y = -6
3y = -6
y = -2 <<< found y !

Substitute y back into either equation…

x - 3y = 3 or 2x = 3y
x - 3(-2) = 3 or 2x = 3(-2)
x + 6 =3 or 2x = -6
x = 3 - 6 or x = -6 / 2
x = -3 or x = -3

Found x !

So the solution to our system is
[ x = -3 ; y = - 2 ]
——————————

NOTICE : With two (2) equations and two (2) variables we have four (4) ways we can start:
First equation by first variable
First equation by second variable
Second equation by first variable
Second equation by second variable

The number of ways we can start is
(# of equations) * (# of variables)

The reason we only had one way to do the first example was because the first equation in that example was trivial. We really only had one real equation and one real variable.

I hope that helps.

Michael

 

Answer 2

y=-6  Y=-2x-2 

 

substitute -6 for y

-6=-2x-2  multiply by -1 both sides

6=2x+2

4=2x

2=x 

 

 second system is there a typo? 

 

X-3y=3

2x-3y=0 since they are equal subtracting them is zero

              two equations two un-knows

x-3y = 3

-2x+3y=0  mult by -1  and add equations

 

-x= 3

x= -3 

solve top equation by substitution

-3-3y=3

y=-2

solution x=-3 , y=-2