x^2 + 9y^2 = 37
x - 2y = -3
To find the solution to the given system of equations, you can use the method of substitution. Here's how to do it:
Step 1: Solve one of the equations for one variable in terms of the other variable. Let's solve the second equation, x - 2y = -3, for x:
x = 2y - 3
Step 2: Substitute the expression for x (from Step 1) into the other equation and solve for y. In this case, substitute x = 2y - 3 into the first equation:
(2y - 3)^2 + 9y^2 = 37
4y^2 - 12y + 9 + 9y^2 = 37
13y^2 - 12y - 28 = 0
Step 3: Solve the resulting quadratic equation for y. You can use factoring, completing the square, or the quadratic formula to find the values of y. In this case, let's use factoring:
(13y + 14)(y - 2) = 0
Setting each factor equal to zero gives two possible values for y:
13y + 14 = 0 -> y = -14/13
y - 2 = 0 -> y = 2
Step 4: Substitute the values of y back into the second equation to find the corresponding values of x. Using y = -14/13, we have:
x - 2(-14/13) = -3
x + 28/13 = -3
x = -3 - 28/13
x = -67/13
Using y = 2, we have:
x - 2(2) = -3
x - 4 = -3
x = -3 + 4
x = 1
Step 5: Therefore, the solutions to the system of equations are x = -67/13 and y = -14/13, and also x = 1 and y = 2.