Solve the following system of equations using the elimination method.%0D%0A%0D%0A2x+y=3.5%0D%0A2%0D%0A%0D%0A+%0D%0A%0D%0A=%0D%0A3.5%0D%0A, −x+2y=2.5
To solve the system of equations using the elimination method, we need to eliminate one variable by adding or subtracting equations.
Given equations:
2x + y = 3.5 ----(1)
−x + 2y = 2.5 ----(2)
To eliminate x, we'll multiply equation (1) by 2 and equation (2) by 2.
2(2x + y) = 2(3.5)
−2(−x + 2y) = 2(2.5)
Simplifying these equations, we get:
4x + 2y = 7 ----(3)
2x - 4y = 5 ----(4)
Now, we will add equations (3) and (4) to eliminate the variable x:
(4x + 2y) + (2x - 4y) = 7 + 5
6x - 2y = 12
To solve for x, we'll divide both sides of the equation by 6:
(6x - 2y)/6 = 12/6
x - (1/3)y = 2 ----(5)
Now, we will solve for y. We'll multiply equation (2) by 2 and add it to equation (1) to eliminate the variable y:
2(−x + 2y) + (2x + y) = 2(2.5) + 3.5
−2x + 4y + 2x + y = 5 + 3.5
5y = 8.5
To solve for y, we'll divide both sides of the equation by 5:
5y/5 = 8.5/5
y = 1.7
Now, substitute the value of y back into equation (5) to solve for x:
x - (1/3)(1.7) = 2
x - 0.57 = 2
x = 2 + 0.57
x = 2.57
Therefore, the solution to the system of equations is x = 2.57 and y = 1.7.