x^2 + 9y^2 = 37
x - 2y = -3
simultaneous equation by elimination
elimination is usually done with linear equations. Using substitution,
x = 2y-3, so we have
(2y-3)^2 + 9y^2 = 37
4y^2 - 12y + 9 + 9y^2 = 37
13y^2 - 12y - 28 = 0
(y-2)(13y+14) = 0
(9y2+x2y)(9y2-x2y)
To solve this system of equations by elimination, we want to eliminate either the x or y variable by manipulating the two equations. Let's eliminate the x variable in this case.
We have two equations:
1) x^2 + 9y^2 = 37
2) x - 2y = -3
To eliminate the x variable, we can multiply both sides of equation 2) by x:
2x - 4y = -6
Now, we have:
1) x^2 + 9y^2 = 37
2) 2x - 4y = -6
Next, we want to manipulate the equations so that the coefficients of the x terms will be the same with opposite signs, or the coefficients of the y terms will be the same with opposite signs.
In this case, let's focus on the coefficients of the x terms. We have x^2 and 2x. We can multiply equation 1) by 2:
2(x^2 + 9y^2) = 2(37)
2x^2 + 18y^2 = 74
Now, we have:
3) 2x^2 + 18y^2 = 74
4) 2x - 4y = -6
Notice that equations 3) and 4) now have the same coefficient (2) for the x variable.
By subtracting equation 4) from equation 3), the x term will be eliminated:
(2x^2 + 18y^2) - (2x - 4y) = 74 - (-6)
2x^2 + 18y^2 - 2x + 4y = 80
Simplifying further:
2x^2 - 2x + 18y^2 + 4y = 80
At this point, we have an equation that only contains the y variable. We can solve for y using this equation.