Solve the following system algebraically. Verify the solution
y=(1/2)(x-3)^2+4
y=-(1/2)(x-1)^2+6
just equate the two expressions, since the y needs to be the same
(1/2)(x-3)^2+4 = -(1/2)(x-1)^2+6
(x-3)^2 + 8 = -(x-1)^2 + 12
x^2 - 6x + 9 + 8 = -x^2 + 2x - 1 + 12
2x^2 -8x +6 = 0
x^2-4x+3 = 0
(x-1)(x-3) = 0
To solve the system algebraically, we can set the two equations equal to each other:
(1/2)(x-3)^2 + 4 = -(1/2)(x-1)^2 + 6
First, let's simplify the equations:
Multiply both sides of the first equation by 2 to eliminate the fraction:
(x-3)^2 + 8 = -(x-1)^2 + 12
Expand both squares:
(x^2 - 6x + 9) + 8 = (-x^2 + 2x - 1) + 12
Now, combine like terms on both sides:
x^2 - 6x + 17 = -x^2 + 2x + 11
Add x^2 and subtract 2x from both sides:
2x^2 - 8x + 17 = 11
Subtract 11 from both sides:
2x^2 - 8x + 6 = 0
Now, we can solve this quadratic equation. We can either factor it or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 2, b = -8, and c = 6, so substituting these values into the formula:
x = (-(-8) ± √((-8)^2 - 4(2)(6))) / (2(2))
x = (8 ± √(64 - 48)) / 4
x = (8 ± √16) / 4
x = (8 ± 4) / 4
Now, we have two possible values for x:
x = (8 + 4) / 4 = 3
x = (8 - 4) / 4 = 1
To find the corresponding values of y, substitute these x values into either of the original equations.
For x = 3:
y = (1/2)(3-3)^2 + 4
y = (1/2)(0)^2 + 4
y = 0 + 4
y = 4
For x = 1:
y = -(1/2)(1-1)^2 + 6
y = -(1/2)(0)^2 + 6
y = -(1/2)(0) + 6
y = 0 + 6
y = 6
Therefore, the solution to the system of equations is x = 1, y = 6 and x = 3, y = 4.
To solve the system algebraically, we need to find the values of x and y that satisfy both equations simultaneously.
Step 1: Setting the two equations equal to each other:
(1/2)(x-3)^2 + 4 = -(1/2)(x-1)^2 + 6
Step 2: Remove the parentheses by squaring both binomials:
(1/2)(x^2 - 6x + 9) + 4 = -(1/2)(x^2 - 2x + 1) + 6
Step 3: Distribute the 1/2 to each term inside the parentheses:
(1/2)x^2 - 3x + (9/2) + 4 = -(1/2)x^2 + x - (1/2) + 6
Step 4: Combine like terms:
(1/2)x^2 - (1/2)x^2 - 3x - x + 9/2 + 4 - 1/2 - 6 = 0
Step 5: Simplify the equation:
-4x + (9/2) + (7/2) - (13/2) = 0
Step 6: Combine like terms:
-4x + 3 = 0
Step 7: Solve for x:
-4x = -3
x = (-3)/(-4)
x = 3/4
Step 8: Substitute the value of x into one of the original equations to find y:
y = (1/2)(3/4 - 3)^2 + 4
y = (1/2)(3/4 - 12/4)^2 + 4
y = (1/2)(-9/4)^2 + 4
y = (1/2)(81/16) + 4
y = 81/32 + 4
y = 81/32 + 128/32
y = 209/32
Therefore, the solution to the system of equations is x = 3/4 and y = 209/32.