Identify the solution(s) of the system of equations, if any.
X + 2y = 4
9x – 4y = 3
a) (1, 2/3)
b) (1, 3/2)
c) (2/3, 1)
d) (3/2,1)
2 x + 4 y = 8
9 x - 4 y = 3
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11 x = 11
x = 1
y = 3/2
b) (1, 3/2)
To identify the solution(s) of the system of equations, we can solve the system by a method such as substitution or elimination. Let's use the elimination method to find the solution.
Given system of equations:
1) X + 2y = 4
2) 9x - 4y = 3
To eliminate one of the variables (either x or y), we need to make the coefficients of one of the variables the same in both equations. In this case, if we multiply equation 1 by 4, and equation 2 by 2, the coefficients of y will become the same.
Multiply equation 1 by 4:
4(X + 2y) = 4(4)
4X + 8y = 16
Multiply equation 2 by 2:
2(9x - 4y) = 2(3)
18x - 8y = 6
Now, we can eliminate y by subtracting the second equation from the first equation:
(4X + 8y) - (18x - 8y) = 16 - 6
4X + 8y - 18x + 8y = 10
-14x + 16y = 10
Simplify:
-14x + 16y = 10 --> -7x + 8y = 5
Now, we have a new equation:
-7x + 8y = 5 (equation 3)
To solve for x, we need to eliminate y again. For this, we can multiply equation 1 by 9 and equation 3 by 1:
9(X + 2y) = 9(4)
9X + 18y = 36
1(-7x + 8y) = 1(5)
-7x + 8y = 5
Now, we can eliminate y by subtracting the second equation from the first equation:
(9X + 18y) - (-7x + 8y) = 36 - 5
9X + 18y + 7x - 8y = 31
16x + 10y = 31
Simplify:
16x + 10y = 31 (equation 4)
Now, we need to solve equations 3 and 4 simultaneously. We can do this by multiplying equation 3 by 10 and equation 4 by 8:
10(-7x + 8y) = 10(5)
-70x + 80y = 50
8(16x + 10y) = 8(31)
128x + 80y = 248
By subtracting these two equations, we can eliminate y:
(-70x + 80y) - (128x + 80y) = 50 - 248
-70x + 80y - 128x - 80y = -198
-198x = -198
x = -198 / -198
x = 1
Now that we have the value of x, we can substitute it into equation 3 or 4 to find the value of y. Let's use equation 3:
-7x + 8y = 5
-7(1) + 8y = 5
-7 + 8y = 5
8y = 5 + 7
8y = 12
y = 12 / 8
y = 3 / 2
So, the solution to the system of equations is (x, y) = (1, 3/2).
Therefore, the correct answer is option d) (1, 3/2).