Solve the system of equations.
y = 2x² – 3
y = 3x – 1
A. no solution
B. (–one-half, 5), (2, –start fraction 5 over 2 end fraction)
C. (–one-half, –start fraction 5 over 2 end fraction), (2, 5)
D. (one-half, start fraction 5 over 2 end fraction), (2, 5)
AAAaannndd the bot gets it wrong yet again!
2x² – 3 = 3x – 1
2x² - 3x - 2 = 0
(2x+1)(x-2) = 0
x = -1/2, 2
I apologize for my previous response, and thank you for bringing it to my attention. You are correct that the system of equations has solutions. Solving for x by factoring, we get:
2x² - 3x - 2 = 0
(2x + 1)(x - 2) = 0
x = -1/2, 2
Substituting each of these values of x back into either of the original equations, we can find the corresponding values of y:
For x = -1/2: y = 2(-1/2)² - 3 = -2
For x = 2: y = 2(2)² - 3 = 5
Therefore, the solutions to the system of equations are (x, y) = (-1/2, -2) and (x, y) = (2, 5). The answer is (D) (one-half, 5), (2, 5).
To solve the system of equations, we need to find the values of x and y that satisfy both equations simultaneously.
Since both equations are already in the form y = ..., we can set them equal to each other:
2x² - 3 = 3x - 1
Rearranging the equation to standard form, we have:
2x² - 3x - 1 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Plugging in the values from our equation, we have:
x = (3 ± √(3² - 4(2)(-1))) / (2(2))
Simplifying within the square root:
x = (3 ± √(9 + 8)) / 4
x = (3 ± √17) / 4
So the values of x are (3 + √17) / 4 and (3 - √17) / 4.
Now, substitute these values back into either equation to find the corresponding y-values. Let's use the second equation:
For x = (3 + √17) / 4:
y = 3( (3 + √17) / 4 ) - 1
y = (9 + 3√17)/4 - 4/4
y = (9 + 3√17 - 4) / 4
y = (5 + 3√17) / 4
So one set of solutions is ( (3 + √17) / 4 , (5 + 3√17) / 4 ).
For x = (3 - √17) / 4:
y = 3( (3 - √17) / 4 ) - 1
y = (9 - 3√17)/4 - 4/4
y = (9 - 3√17 - 4) / 4
y = (5 - 3√17) / 4
So the other set of solutions is ( (3 - √17) / 4 , (5 - 3√17) / 4 ).
Therefore, the correct answer is option D. ( (3 + √17) / 4 , (5 + 3√17) / 4 ) and ( (3 - √17) / 4 , (5 - 3√17) / 4 ).
We can set the two expressions for y equal to each other since they both equal y:
2x² – 3 = 3x – 1
Simplifying this equation, we get:
2x² - 3x + 2 = 0
We can then solve using the quadratic formula:
x = (-(-3) ± sqrt((-3)^2 - 4(2)(2))) / 2(2)
x = (3 ± sqrt(9 - 16)) / 4
x = (3 ± sqrt(-7)) / 4
Since we cannot take the square root of a negative number in real numbers, there are no real solutions to this system of equations, so the answer is (A) no solution.