Question

How do you solve this system of equations using the substitution method: `2x + y = 13; 5x - 2y = 10`?

Answer 1

See explanation.

Explanation:

The substitution method is a way to solve the system of equations which requires calculating one of the variables from one equation and substituting it into the other equation.

The initial system is:

`{(2x+y=13),(5x-2y=10):}` ##

You can easily calculate `y` from the first equation:

`y=-2x+13`

Now if you substitute this value in the second equation you get the equation with one variable only (`x`):

`5x-2*(-2x+13)=10`

`5x+4x-26=10`

`9x=36 =>x=4`

Now you can substitute the calculated value of `x` into any of the equations:

`y=-2*4+13=>y=-8+13 =>y=5`

Finally we can write the answer:

This system has one solution:

`{(x=4),(y=5):}`

Answer 2

`(x,y)to(4,5)`

Explanation:

`2x+y=13to(1)`

`5x-2y=10to(2)`

`"rearrange equation "(1)" to give y in terms of x"`

`rArry=13-2xto(3)`

`color(blue)"substitute "y=13-2x" into equation "(2)`

`rArr5x-2(13-2x)=10`

`rArr5x-26+4x=10`

`rArr9x-26=10`

`"add "26" to both sides"`

`9xcancel(-26)cancel(+26)=10+26`

`rArr9x=36`

`"divide both sides by "9`

`(cancel(9) x)/cancel(9)=36/9`

`rArrx=4`

`"substitute this value into equation "(3)" and evaluate for y"`

`rArry=13-(2xx4)=13-8=5`

`color(blue)"As a check"`

`"substitute these values into equation "(2)`

`(5xx4)-(2xx5)=20-10=10larrcolor(blue)"True"`

`rArr"point of intersection "=(4,5)`
graph{(y+2x-13)(y-5/2x+5)=0 [-10, 10, -5, 5]}

Answer 3

Solution: `x=4 , y=5`

Explanation:

`2x+y=13 ; (1) , 5x-2y=10; (2)`. From equation (1)

we get `y=13-2x`. Substituting `y=13-2x` in

equation (2) we get `5x - 2(13-2x)=10` or

`5x - 26+4x =10 or 9x = 10+26 or 9x=36 or x=4`.

Substituting `x=4` in equation (1) we get ,

`2*4+y=13 :. y = 13-8 or y=5 :. x=4 , y=5`

Solution: `x=4 , y=5` [Ans]