Solve the system of equations.
y=2x^2 -3
y=3x-1
a. no solution
b. (-1/2, 5)(2, -5/2)
c. (-1/2,-5/2) (2,5)***
d. (1/2,5/2) (2,5)
By substitution or the comparison method set equation 1 equal to equation 2 and solve for x
2x^2-3 = 3x -1 then sub your answer for x back into equation 2 and solve for y : )
your indicated answer is correct.
To solve the system of equations, let's set the two equations equal to each other:
2x^2 - 3 = 3x - 1
Rearranging the equation, we get:
2x^2 - 3x + 2 = 0
Now, we can factor the quadratic equation. Looking at the last term (2) and the middle term (-3x), we need to find two numbers that multiply to give 2 and add up to -3. The numbers -1 and -2 fulfill these requirements:
(2x - 1)(x - 2) = 0
Setting each factor equal to zero, we can solve for x:
2x - 1 = 0 => x = 1/2
x - 2 = 0 => x = 2
Now that we have the values of x, we can substitute them back into either of the original equations to find the corresponding y-values. Let's use the first equation:
y = 2x^2 - 3
When x = 1/2:
y = 2(1/2)^2 - 3
y = 2(1/4) - 3
y = 1/2 - 3
y = -5/2
When x = 2:
y = 2(2)^2 - 3
y = 2(4) - 3
y = 8 - 3
y = 5
Therefore, the solution to the system of equations is (x, y) = (-1/2, -5/2) and (2, 5).
So the correct answer is c. (-1/2,-5/2) (2,5).
To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution.
Step 1: We are given two equations:
1. y = 2x^2 - 3 ---(Equation 1)
2. y = 3x - 1 ---(Equation 2)
Step 2: Since both equations are equal to y, we can set them equal to each other:
2x^2 - 3 = 3x - 1
Step 3: Rearrange the equation to put it in standard quadratic form:
2x^2 - 3x - 2 = 0
Step 4: To solve this quadratic equation, we can factor it or apply the quadratic formula. Let's factor it:
(2x + 1)(x - 2) = 0
Step 5: Set each factor equal to zero and solve for x:
2x + 1 = 0 ---> 2x = -1 ---> x = -1/2
x - 2 = 0 ---> x = 2
So, the possible values for x are x = -1/2 and x = 2.
Step 6: Now, substitute the found values of x back into one of the original equations (Equation 1 or Equation 2) to find the corresponding values of y.
For x = -1/2:
Using Equation 1: y = 2(-1/2)^2 - 3 ---> y = 2(1/4) - 3 ---> y = 1/2 - 3 ---> y = -5/2
For x = 2:
Using Equation 1: y = 2(2)^2 - 3 ---> y = 2(4) - 3 ---> y = 8 - 3 ---> y = 5
So, the solutions to the system of equations are:
(-1/2, -5/2) and (2, 5)
Therefore, the correct option is c. (-1/2, -5/2) (2, 5).