4xsquare-9ysquare=19,
2x+3y=1
4x^2 - 9y^2 = 19
(2x-3y)(2x+3y) = 19
2x-3y = 19
Not sure where you want to go with that.
2 x + 3 y = 1
Subtract 2 x to both sides
3 y = 1 - 2 x
Divide both sides by 3
y = ( 1 - 2 x ) / 3
Plug this into: 4 x² - 9 y² = 19
4 x² - 9 [ ( 1 - 2 x ) / 3 ]² = 19
4 x² - 9 ( 1 - 2 x )² / 3² = 19
4 x² - 9 [ 1² - 2 ∙ 1 ∙ 2 x + ( 2 x )² ] / 9 = 19
4 x² - ( 1 - 4 x + 4 x² ) = 19
4 x² - 1 + 4 x - 4 x² = 19
- 1 + 4 x = 19
Add 1 to both sides
4 x = 20
Divide both sides by 4
x = 5
Plug this into: y = ( 1 - 2 x ) / 3
y = ( 1 - 2 ∙ 5 ) / 3
y = ( 1 - 10 ) / 3
y = - 9 / 3
y = - 3
Final solution:
x = 5 , y = - 3
Proof:
4 x² - 9 y² = 19
4 ( 5 )² - 9 ( - 3 )² = 19
4 ∙ 25 - 9 ∙ 9 = 19
100 - 81 = 19
19 = 19
2 x + 3 y = 1
2 ∙ 5 + 3 ∙ ( - 3 ) = 1
10 - 9 = 1
1 = 1
4x^2 - 9y^2 = 19 and 2x+3y = 1
the first one factors to
(2x+3y)(2x-3y) = 19
(1)(2x-3y) = 19
2x-3y = 19 with 2x+3y = 1
add them
4x=20
x=5
then 10 + 3y = 1
y = -3
https://www.wolframalpha.com/input/?i=solve+4x%5E2+-+9y%5E2+%3D+19+%2C+2x%2B3y+%3D+1
To solve this system of equations, we can use the method of substitution.
Let's solve the second equation for x in terms of y:
2x + 3y = 1
2x = 1 - 3y
x = (1 - 3y) / 2
Now we substitute this value of x into the first equation:
4x^2 - 9y^2 = 19
4((1 - 3y) / 2)^2 - 9y^2 = 19
(1 - 3y)^2 - (9/4)y^2 = 19
Expanding the equation:
(1 - 6y + 9y^2) - (9/4)y^2 = 19
1 - 6y + 9y^2 - (9/4)y^2 = 19
1 - 6y + 9y^2 - 9y^2/4 = 19
Combining like terms:
(9/4 - 6)y + (9 - 19) = 0
(9/4 - 6)y - 10 = 0
(9/4 - 24/4)y - 10 = 0
(-15/4)y - 10 = 0
-15y - 40 = 0
-15y = 40
y = -40 / 15
y = -8/3
Now substitute this value of y back into the second equation to solve for x:
2x + 3(-8/3) = 1
2x - 8 = 1
2x = 9
x = 9/2
Therefore, the solution to the system of equations is x = 9/2 and y = -8/3.