Solving by the substitution method!!!?

Question

I need some serious help, I am not understanding the following problems. Please give an explanation of how I am suppose to solve the following equations.
1) x = y +3 and 3x -2y =4

2) 2x - y = 4 and 2x - y = 3

3) x + 3y = 2 and -x + y =1

Please show work so that I can better understand.

Thank you!

Answer 1

1)Take the first equation and substitute it for the second equation:
3(y+3)-2y=4
3y+9-2y=4
y+9=4
y=-5
x=-5+3
x=-2
(-2,-5)

Check:
-2=-5+3
-2=-2

3(-2)-2(-5)=4
-6+10=4
4=4

2)For the first equation, subtract 2x from both sides:
-y=-2x+4
Divide -1 from both sides:
y=2x-4
Put that equation into the second equation:
2x-(2x-4)=3
2x-2x+4=3
0x+4=3
0x=-1
No Solution

3)On the second equation, add x from both sides:
y=x+1
Substitute it into the first equation:
x+3(x+1)=2
x+3x+3=2
4x+3=2
4x=-1
x=-1/4
-(-1/4)+y=1
1/4+y=1
1/4-1/4+y=4/4-1/4
y=3/4
(-1/4,3/4)

Check:
-1/4+3(3/4)=2
-1/4+9/4=2
8/4=2
2=2

-(-1/4)+3/4=1
1/4+3/4=1
1=1

Answer 2

1)
x=y+3 (1) and 3x-2y=4 (2)
now sub the first equation into the second as follow:
3(y+3) - 2y = 4. Then distribute 3 into the paranthesis
3y + 9 - 2y = 4. Then simplify
9 + y = 4. Subtract 9 from both side to isolate y
9-9 + y = 4 - 9 => y = -5. Then pluck the value of y back into the first equation to solve for x
x = -5 +3 = -2

2)
2x - y = 4 and 2x- y = 3. First, you need to look at these two equation, and you see that the left hand side of these equation (2x-y) are the same; however, the right side of these equation are not equal. This is like saying that you are 4 year-old and at the same time you are 3 year-old. You simply just can't have both statements true at same time. So, back to the problem, there is no solution for these equation.

3)
x + 3y = 2 and -x + y = 1

first, solve for x in the first equation by isolate it on 1 side

x + 3y - 3y = 2 - 3y => x= 2- 3y. Next sub the right side of this equation into the second equation where it say x

-x + y = 1 => -(2 - 3y) + y = 1. Then integrate the negative inside the paranthesis.

-2 + 3y + y = 1. Then simplify it.

-2 + 2 + 4y = 1 + 2 => y=3/4. then sub y back to the first equation where-ever there y is

x + 3 y = 2 => x + 3*(3/4) = 2. then simplify it
x + 9/4 = 2 => x = 2 - 9/4 = 8/4 - 9/4 = -1/4

Answer 3

Hiya.
In this type of problem, we use what is called simultaneous equations to solve. This will result in both an x and y value (ie what we are looking for.)

1. In the first problem you are given x = y + 3 and 3x - 2y = 4.
To solve this you need to have both equations in the same form, i.e. have the x values and the y values at one side and the constants on the other side. Therefore we change the first equation from x = y + 3 to x - y = 3 (this is done simply by bringing the y to the other side of the equals to sign)

Now both equations are in the same form!
x - y = 3
3x - 2y = 4

To get the first value, we need to be able to cancel out one of the "letter values" (i.e. either the x or y) This can be done by multiplying ALL of the first equation by 2, to give 2x - 2y = 6.
Now the two equations are: 2x - 2y = 6
3x - 2y = 4

To cancel the y's, we need to change the sign on one equation, so for arguments sake, we will change the sign on equation 2, to give: 2x - 2y = 6
-3x + 2y = -4
(This is done by multiplying across all of equation two by minus 1)

Now seeing as -2y + 2y = o, the y's are cancelled.
This leaves us with 2x = 6
-3x = -4

Dealing with the x's, we get 2x - 3x and this is then equalled to whats on the other side 6 - 4.

Therefore 2x - 3x = 6 - 4
-x = 2
x = -2

Now we have the value of x, we put this value into any of the above equations to find the corresponding y value.
So putting this into our original equation (x = y + 3), we get:

-2 = y + 3
-2 -3 = y
-5 = y

So the answer to problem 1 is x = -2 and y = -5



2. For the 2nd problem, the following is the solution (explanation same as problem 1)

2x - y = 4
2x - y = 3

Changing signs on the bottom equation we get,

2x - y = 4
-2x + y = -3

Y's now cancel, leaving

2x - 2x = 4 - 3
0x = 1
x = 1 / 0 (not possible in mathematics to divide by zero,
therefore there is no x value)

Hence if we have no x value, there can be no y value.

Conclusion: The above equations cannot be solved!




3. For the final problem, the following is the solution.

x + 3y = 2
-x + y = 1

Straight away the x's cancel out each other (as x - x = 0) leaving behind

3y + y = 2 + 1
4y = 3
y = 3/4

If y = 3/4, the corresponding x value is found by putting this back into one of the above equations!

-x + y = 1
-x + 3/4 = 1
-x = 1 - 3/4
-x = 1/4
x = -1/4

Hope you understand it now!

Answer 4

1) x = y +3 and 3x -2y =4
Substitute y+3 for x in the 2nd equation getting:
3(y+3) - 2y =4
3y+9 -2y =4
y+9 =4
y=4-9 =-5
x= y+3 = -5+3 = -2


2) 2x - y = 4 and 2x - y = 3
THis equation cannot be solved. It is impossible for 2x-y to = both 4 and 3. That would require 4=3 which of course is not true.

3) x + 3y = 2 and -x + y =1
Add the two equations getting
4y =3 so y=3/4
x+3(3/4) =2
x=2-9/4= -1/4

Answer 5

1)
x = y + 3
3x - 2y = 4

Substitute x in the second equation:
3(y + 3) - 2y = 4
3y + 9 - 2y = 4
3y - 2y = 4 - 9
y = -5

Now substitute the y in the first equation:
x = -5 + 3
x = -2

2)
2x - y = 4
2x - y = 3

I think you have a typo in here: 2x - y cannot be equal to 4 and 3 at the same time.

3)
x + 3y = 2
-x + y = 1

Modify the first equation to help you substitute x:
x = 2 - 3y
Now you can do the same thing as in the 1st problem:
Substitute x in the second equation:
- (2 - 3y) + y = 1
-2 + 3y + y = 1
3y + y = 1 + 2
4y = 3
y = 3/4

Substitute y in the first equation:
x = 2 - 3 (3/4)
x = 2 - 9/4
x = -1/4

Answer 6

1. First off you have to solve for one specific variable. I chose x since it was already alone in one.

x=y+3 3x-2y=4
3x=4+2y
x=4/3+2/3y

What I did above was first I moved the 2y over the = and make it positive. Then I divided the 3 to both sides so that I got rid of the 3 with the x.

The next step is to set your equations to = each other.

y+3=4/3+2/3y

After that you solve for y.

y=-5/3+2/3y
1/3y=-5/3
y=-5

What I did above was first I got rid of the 3 in y+3 by subtracting it to 4/3. This gave me -5/3, which is where I got y=-5/3+2/3y. Then I said 1 minus 2/3 which gave me 1/3. So that is where I got 1/3y=-5/3. So then I divided -5/3 and1/3 together to get y alone. with doing this it gave me -5.

After you get what y equals. Then you have to plug into the original to find out what x is.

You only have to do it to one. I always pick the easiest looking one.So I have picked x=y+3 to use.

x=y+3
x= -5+3
x=-2

What I did was I pluged -5 into the spot where y is and after that all that is left is an addition problem.

So your ordered pair is (-2,-5).

2. 2x-y=4 2x-y=3
-y=4-2x -y=3-2x
y=-4+2x y=-3+2x 2(-1)-y=4
-2-y=4
-y=6
y=6

-4+2x=-3+2x
-1+2x=2x
-1=x (-1,6)

3. x+3y=2 -x+y=1
x=2-3y -x=1-y
x=-1+y

2-3y=-1+y -x+3/4=1
2=-1+4y -x=1/4
3=4y x=-1/4
3/4=y

(-1/4,3/4)

Answer 7

Yes, I'm doing it now and it takes a while to type it up

When you have 2 equations and 2 unknowns, the equations are called simultaneous, because they are true at the same time. You use this information to solve for x and y using the 2 equations.

x = y + 3

3x - 2y = 4...........You have been told to use the substitution method, so you want to choose the simpler of the 2 equations to start.

No competition....the top equation contains 1x and 1y, which is easier than dealing with 3s and 2s.

You already know that x = y + 3.............and because these equations are true at the same time, everywhere x appears in the second equation, you can write (y + 3).

I have put this in a bracket, as we need to remember to multiply the whole expression in the next part.

3(y + 3) - 2y = 4

3y + 9 - 2y = 4

y + 9 = 4.............subtract 9 from each side to get y alone on one side

y = 4-9

y = -5................now take this value of y back to the first equation (everywhere you see y write -5) and find the value of x

x = -5 + 3

x = -2

Now for a check, take boths values to the second equation and see if substituting these values makes the equation "true"

3(-2) - 2(-5) = 4

-6 + 10 = 4..........remember - times - equals +

Now you can do the others EXCEPT that the second pair are NOT simultaneous equations.

If 2x - y = 4 you can never choose values of x and y that make it possible for 2x-y to equal 3

Good luck and if you don't understand, be sure to ask your teacher to go through it again with you.

Answer 8

x=y+3 and 3x-2y=4

First you do the substitution, that is take the y+3 and put it where the x is in the 3x-2y=4 equation:

3(y+3)- 2y =4
Now you solve for y:
3y+9 -2y =4
y=-5

Now you solve the next two problems in the same way

Answer 9

Substitue 4y-2 or 6y+8 into one of the equations, whichever one you choose will be fine because it will be the same equation no matter what, just backwards. I'll try to simplify it more: eq. 1) x=4y-2 eq. 2) x=6y+8 Substitute either 4y-2 or 6y+8 into the other equation. If you use 4y-2 from eq. 1, eq. 2 would now look like this: 4y-2=6y+8 because if x is 4y-2, you can just put it right into where the x is in eq. 2 - because x IS 4y-2. It is the same for if you use 6y+8 from eq. 2 -- x IS 6y+8 so you can put this phrase into equation one where the x is because 6y+8 equals x. It would look like this - 6y+8=4y-2 Then after you have either 6y+8=4y-2 or 4y-2=6y+8, you can solve for y by adding together like terms. Using 4y-2=6y+8...... Add the 4y and 6y together by subtracting 4y from the positive 4y on the left side of the equal sign and by 6y to get 2y on the right side of the equal sign : 4y-2=6y+8 -4y -4y ^ the +4y and -4y cancels out leaving... -2=2y+8 you are trying to get y by itself, so you need to subtract 8 from the positive 8 on the right side and subtract 8 from the negative 2 on the left to get -10 on the left: -2=2y+8 -8 -8 ^ the +8 and - 8 cancels out leaving.... -10=2y To get y by itself, you need to divide the 2y by 2 (do the opposite to cancel the 2 out, like we did when it was addition/subtraction), and divide the -10 by 2: -10=2y -10/2=2y/2 ^the multiplied 2 and divided 2 cancels out leaving.... -5=y <--- -5 IS y ---- So now that you have y, you can put in -5 in either eq.1 or 2 and solve : (using eq 1): x = 4y-2 x = 4(-5)-2 x = -20-2 x = -22 ! So the ordered pair to solve these equations is (x, y) : (-22, -5)

Answer 10

1) since x=y+3
3x - 2y =4 can be written as 3(y+3) - 2y = 4
then, y +9=4 => y=-5

Solve the others similarly.