To solve the system of equations using the elimination method, we aim to eliminate one of the variables by adding or subtracting the equations. Let's start by making the coefficients of either 'x' or 'y' the same in magnitude but opposite in sign.
Let's work with the second equation and multiply it by 2 to match the coefficient of 'x' in the first equation:
Equation 1: -2x + 5y = 19
Equation 2 (multiplied by 2): -10x - 20y = -40
Now, if we add the modified Equation 2 to Equation 1, the 'y' terms will cancel out:
(-2x + 5y) + (-10x - 20y) = 19 - 40
-12x - 15y = -21
Divide the entire equation by -3 to simplify:
4x + 5y = 7
Now we have a new equation with simplified coefficients. Let's solve this equation for one variable (let's solve for 'x'):
4x = 7 - 5y
x = (7 - 5y) / 4
Now we can substitute this value of 'x' into either of the original equations to solve for 'y'. Let's use the first equation:
-2x + 5y = 19
-2((7 - 5y) / 4) + 5y = 19
Simplify and solve for 'y':
-14/4 + 10y/4 + 5y = 19
-14 + 10y + 20y = 76
30y = 90
y = 90 / 30
y = 3
Now that we have the value of 'y', we can substitute it back into the equation for 'x':
x = (7 - 5y) / 4
x = (7 - 5(3)) / 4
x = (7 - 15) / 4
x = -8 / 4
x = -2
So, the solution to the system of equations is:
x = -2
y = 3