solve the simultaneous equation
2x+2y=7
x^2-4y^2=8
x^2-4y=8.............(1)
2x+2y=7
or 2x=7-2y
or x=7-2y/2......(2)
from 1&2
(7-2y)^2-4y=8
or 49-28y+4y^2-4y=8
or 4y^2-32y=41
4y(y-32)=41
either 4y=41....(a)
or y-32=41.....(b)
from a
y=41/4=10.25
from b
y=73
so, y=10.25 or 73
To solve the simultaneous equation 2x + 2y = 7 and x^2 - 4y^2 = 8, we can use the method of substitution. Here's how to solve it step by step:
Step 1: Solve one of the equations for one variable in terms of the other variable. Let's solve the first equation, 2x + 2y = 7, for x:
2x = 7 - 2y
x = (7 - 2y) / 2
Step 2: Substitute this expression for x into the second equation, x^2 - 4y^2 = 8:
[(7 - 2y) / 2]^2 - 4y^2 = 8
Step 3: Simplify the equation by expanding the square and combine like terms:
(49 - 28y + 4y^2) / 4 - 4y^2 = 8
49 - 28y + 4y^2 - 16y^2 = 32
Step 4: Combine the y^2 terms and bring all terms to one side of the equation:
-12y^2 - 28y + 17 = 0
Step 5: Factoring or using the quadratic formula, solve for y. In this case, the equation does not factor nicely, so we will use the quadratic formula: y = [-b ± √(b^2 - 4ac)] / 2a
y = [-(-28) ± √((-28)^2 - 4(-12)(17))] / (2(-12))
y = (28 ± √(784 + 816)) / (-24)
y = (28 ± √1600) / (-24)
y = (28 ± 40) / (-24)
So we have two possible values for y: y = (28 + 40) / (-24) or y = (28 - 40) / (-24)
Simplifying each value gives us y = 1 or y = -1/2.
Step 6: Substitute each value of y back into the equation x = (7 - 2y) / 2 to find the corresponding values of x:
For y = 1, x = (7 - 2(1)) / 2 = 6 / 2 = 3
For y = -1/2, x = (7 - 2(-1/2)) / 2 = 8 / 2 = 4
Therefore, the solution to the simultaneous equation 2x + 2y = 7 and x^2 - 4y^2 = 8 is x = 3 and y = 1, or x = 4 and y = -1/2.