Question

How do you solve the system of equations `3x + 4y = 5` and `9x + 3y = 51`?

Answer 1

I have taken you up to a point where you can take over.

A lot of method detail is given.

Explanation:

As far as I know there are 3 basic approaches. The third uses matrix formation and I will not deal with that. 4 methods if you let software do it for you.

Using the very traditional method of equation adjustment and subtraction.

If you have just 1 equation and only 1 unknown then it is solvable. I will be manipulating to give me that condition.

Note that I will be using the abbreviation of `Eqn` for `Equation`

Given:
`3x+4y=5" "....................Equation(1)`
`9x+3y=51" "....................Equation(2)`

`color(blue)("I chose to eliminate "y" so we determine "x)`

Multiply equation(1) by 4 and equation(1) by 3. This will give the same coefficients for `y`

`9x+12y=15" "....................Equation(1_a)`
`36x+12y=204" "..................Equation(2_a)`

Change the order and apply `Eqn(2_a)-Eqn(1_a)`

`36x+12y=204" "..................Eqn(2_a)`
`ul(color(white)(3)9x+12y=color(white)(2)15)" "..................Eqn(1_a)`
`27x+0color(white)(2y)=189 larr" "Eqn(2_a)-Eqn(1_a)`

`27x=189`

Divide both sides by 27

`color(green)(x+189/27=7)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("I will let you do the next bit")`

Substitute for `x=7` in Eqn(1) or Eqn(2) to determine the value of `y`