1. Solve the System of Equation Algebraically.
6x-3y = -33
2x+y = -1
2. What point in the feasible region maximizes the objective function?
{y-5≤4x
{y+x≤10
{x≥0,y≥3
Maximum for
P=7x-5y
3. What is the result of
(x^4 + 2x^3 + x - 4)/(x - 1)?
4. What are the solutions of the quadratic equation?
-4x^2 + 6x - 3 = 0
1. To solve the system of equations algebraically, we can use the method of substitution or elimination.
Method of Substitution:
From the second equation, we can solve for y:
2x + y = -1
y = -2x - 1
Substitute this value of y into the first equation:
6x - 3(-2x - 1) = -33
6x + 6x + 3 = -33
12x + 3 = -33
12x = -36
x = -3
Substituting this value of x into the second equation:
2(-3) + y = -1
-6 + y = -1
y = 5
Therefore, the solution to the system of equations is x = -3 and y = 5.
2. To find the point in the feasible region that maximizes the objective function, we can graph the inequalities and find the coordinates of the corner point(s) where the lines intersect.
The feasible region is the shaded area where the inequalities are satisfied:
Feasible Region:
{y - 5 ≤ 4x (Equation 1)
{y + x ≤ 10 (Equation 2)
{x ≥ 0, y ≥ 3 (Equation 3)
By graphing these inequalities, we find that the corner point where the lines intersect is (3, 5). This point maximizes the objective function.
3. To simplify the expression (x^4 + 2x^3 + x - 4)/(x - 1), we can use polynomial long division or synthetic division.
Using polynomial long division:
x^3 + 3x^2 + 4x + 5
______________________
x - 1 | x^4 + 2x^3 + 0x^2 + x - 4
- (x^4 - x^3)
_______________
3x^3 + 0x^2 + x
- (3x^3 - 3x^2)
_______________
3x^2 + x - 4
- (3x^2 - 3x)
_______________
4x - 4
- (4x - 4)
_______________
0
Therefore, the simplified expression is x^3 + 3x^2 + 4x + 5.
4. To find the solutions of the quadratic equation -4x^2 + 6x - 3 = 0, we can use the quadratic formula or factoring.
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -4, b = 6, and c = -3.
x = (-6 ± √(6^2 - 4(-4)(-3))) / (2(-4))
x = (-6 ± √(36 - 48)) / (-8)
x = (-6 ± √(-12)) / (-8)
x = (-6 ± 2√3i) / (-8)
x = (3 ± √3i) / 4
Therefore, the solutions to the quadratic equation are x = (3 + √3i) / 4 and x = (3 - √3i) / 4.