Simultaneous equations - i'm missing something, a little help...?

Question

I have two simultaneous equations which I must solve without using trial and improvement:

5x + 3y = 6
3x - 7y = 19

I realise that I need to get the x's or the y's the same by multiplying by something before I can do the next step of adding or subtracting, then finally substitute to get the values.

Could somebody good at maths help me out by working the whole question out with explanations along the way - I can't believe i'm getting caught out, I thought I was doing okay until I came to a certain part.

Answer 1

ok let us eliminate the y term

in order to do that, multiply the top equation by 7 and the bottom by 3. That will cause the y coefficients to be 21 and -21 respectively. When you add the equations, the y terms cancel out and all you are left with is the x terms

7(5x+3y = 6) = 35x + 21y = 42
3(3x-7y = 19) = 9x -21y = 57

adding we get: 44x + 0 = 99
x = 99/44 = 9/4

substituting this x in either equation will get you y

5(9/4) + 3y = 6

11.25 -6 = -3y
y = -1.75

you should substitute into each equation with x and y to see in the equations are true

the other way to do this is to solve for y in the first equation and substitute that y into the second equation and solve for the remaining unknown

from the first equation: y = 6/3-5x/3

y = 2-5x/3

substituting this into the second equation we get

3x -7(2-5x/3) = 19 and solving for x

3x - 14 + 35x/3 = 19

44x/3 = 19 + 14
44x/3 = 33
44x = 99 and x = 9/4 as before

QED

Answer 2

This equation requires the elimination method. You want too cancel out the 5x and 3x and the only answer that can both cancel would be 15. So imagine this: (-3) 5x+3y=6
(+5) 3x-7y=19

See the number in parenthesis above? This shows that you MUST multiply that number by every other number in that equation. This problem has two equations: one is 5x+3y=6 and the other is 3x-7y=19. Once you multiply you should get this
-15x-9y=-18 <::::::::::Top equation
+15x-35y=95 <::::::Bottom equation

Now you made this two part equation easier. Now all you have to do is combine the top equation with the bottom one. What you should get is:

-44y=77 <:::::One equation

Notice now you have one equation then you would have to divide to find Y.

-44y=77

-1 3/4
___
-44/77 Therfore Y equals: -1 3/4
-44
___
33

Since we have found Y we have to find X. In order to do this we replace Y with what Y equals -1 3/4.


3x-7(-1 3/4)=19
3x-49/4=19
3x=19+49/4
3x=19+12 1/4
3x=31 1/4
3x= 125/4
x= 93 3/4

Here is your problem and here is your corrdenance:
( 93 3/4; -1 3/4)

Answer 3

Multiply the first equation by something and the second equation by something so that one of their variables have the same, or opposite, coefficients. Try multiplying the top equation by 3 to get 15x+9y=18; now multiply the bottom equation by 5 to get 15x-35y=95. Next subtract one from the other to eliminate the x variable. For example, subtracting the bottom from the top yields 44y=-77 and y=-77/44 or y=-7/4. Substitute this back into either of the original equations to solve for x.