Simultaneous equations?

Question

please can someone help me with these no matter how much i try, revision guides and other questions do not help me they just confuse me, i have exam in about a week and i realy need to know how to do them, please can someone show me how to do them in the simplest way possible, id really appreciate it,

e.g,

3x + y = 9
2x + y = 7

and

2x + 3y = 10
2x + y = 6

Answer 1

The whole idea behind simultaneous equations is to solve equations with more than one variable, in our case two variables x and y.

eg. 1
equation 1.) 3x + y = 9
equation 2.) 2x + y = 7

First step is to eliminate one of the variables, let's get rid of y.
So what we'll do is multiply equation 2 by negative 1. This gives us:

equation 1.) 3x + y = 9
equation 2.) -2x - y = -7

Now add both equations together. 3x + (-2x) = x , y + (-y) = 0 , and 9 + (-7) = 2.
Our y variable cancels out just like we planned, we are left with a new equation that reads:
x = 2

We now have an x value (2), so plug that x value into one of the two equations (doesn't matter which one) from our problem and solve for y.
equation 1.) 3(2) + y = 9 , y = 3


eg. 2

equation 1.) 2x + 3y = 10
equation 2.) 2x + y = 6

Lets multiply equation 2 by negative 1

equation 1.) 2x + 3y = 10
equation 2.) -2x - y = -6

Now add both equations together: 2x + (-2x) = 0 , 3y + (-y) = 2y and 10 + (-6) = 4

Your left with an equation that reads: 2y = 4 which yields y = 2
Plug this value into one of the two equations from our problem.

equation 2.) 2x +(2) = 6 , x= 2

Basically, you get rid of one variable, solve for the other, then plug back in to get the value of the variable you got rid of.

Answer 2

Example 1. Try to eliminate one of them, either x or y.
3x+y=9 ---- (1)
2x+y=7 ---- (2)
to eliminate y, you must subtract (2) from (1)
this gives x=2 (the y's cancel out)
Now that you know x, you can substitue x in (1)
3(2)+y=9
6+y=9
y=3
You can see that x=2 and y=3 also satisfies the second equation 2x+y=7
If you're having trouble with adding and subtracting variables, you should consider taking a course.
To solve 2x+3y=10 ---- (1)
2x+y=6 ----- (2)
Multiply (2) by 3
This gives 2x+3y=10
6x+3y=18
I'll leave the rest up to you.

Answer 3

3x + y = 9 //equation 1
2x + y = 7 //equation 2

rearrange equation 1 to solve for y
y = 9 -3x //equation 1 rearranged.
now replace y in equation 2 with the 9-3x from equation 1
2x + (9-3x) =7 // substitution in equation 2
now collect terms in equation 2
-x = -2
or
x = 2
with x =2 , put that back into equation 1 and solve for y
y = 9-3(2)
y = 9-6=3
therefore the solution is x = 2, y =3 or (x,y)=(2,3)

similar work required for the second set of equations:

2x + 3y = 10 // eqn1
2x + y = 6 //eqn 2
y = (1/3)(10-2x) // eqn1 rearranged.
2x+(1/3)(10-2x)= 6 // subst into eqn2
6x +10-2x = 18 // multiply both sides by 3 to get rid of (1/3)
4x =8 // collecting
x =2
2(2)+3y=10 // replacing x with 2 in eqn 1
3y=6 //collecting
y=2

therefore (x,y)=(2,2)

Answer 4

Subtract the second equation from the first equation:

3x-2x +y-y = 9-7
x = 2

put that back into the either equation:

3(2) + y = 9
6 + y = 9
y = 3


For the second one, same thing

2x-2x +3y-y = 10-6
2y = 4
y = 2

2x + 2 = 6
2x = 4
x = 2

Answer 5

3x + y = 9
2x + y = 7

Subtract the second equation from the first equation....
3x + y = 9
- 2x + y = 7
-------------------
x = 2

Plug this into either of the first two equations...
3(2) + y = 9
y = 3


2x + 3y = 10
2x + y = 6

Subtract the second equation from the first...
2x + 3y = 10
- 2x + y = 6
---------------------
2y = 4
y = 2

Plug this into either of the first two equations...

2x + 2 = 6
2x = 4
x= 2

Answer 6

3x+y= 9
2x+y= 7
change the signs on the bottom line
-2x-y= -7

3x+y= 9
-2x-y= -7
which equals to 1x= 2
therefore
x= 2

now fill in the x to get y

2x+y= 7
2(2)+y= 7
4+y=7
y= 7-4
y= 3
answer is (2,3)

next sum
2x+3y= 10
2x+y= 6

-2x-y= -6

2x+3y= 10
-2x-y -6
equals
2y= 4
y= 2

this time fill in for y to get x

2x+3y= 10
2x+3(2)=10
2x= 10
x= 5
your answer is (5,2)

if u still dont get it write back to me and i will explain
good luck

Answer 7

Question 1
3x + y = 9
- 2x - y = - 7-----ADD
x = 2
6 + y = 9
y = 3
Solution is x = 2, y = 3

Question 2
2x + 3y = 10
-2x - y = - 6----ADD
2y = 4
y = 2
2x + 6 = 10
x = 2
Solution is x = 2 , y = 2

Answer 8

ok here goes... i m in geometry so i might not know what i m talking about, ok? but its simple, i just plug in numbers, but its
3x+y=9, so x=2, and y=3
2x+3y=10, so x and y=2