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SOLVED PROBLEMS
6.1 Find an equation of the line passing through (2, 5) and (7, 9).
SOLUTION
The general equation of a line is y = a x + b. Substituting the coordinates of the given points leads to the equations 2 a + b = 5 and 7a + b = 9. Solve[2 a + b 5 && 7 a + b 9] 4 {{a 5 ,b 17}} 5 The line has equation y = 4 x + 17 . 5 5
6.2 Find an equation of the circle passing through (1, 4), (2, 7), and (4, 11).
SOLUTION
The general equation of a circle is x2 + y2 + ax + by + c = 0. We substitute the coordinates of the given points into the equation to obtain 17 + a + 4 b + c = 0, 53 + 2 a + 7 b + c = 0, and 137 + 4 a + 11 b + c = 0.
Equations
Solve[{17 + a + 4 b + c 0, 53 + 2 a + 7 b + c 0,137 + 4 a + 11 b + c 0}] {{a 54, b 6, c 13}} The equation of the circle is x2 + y2 54 x + 6 y + 13 = 0.
6.3 Solve the equation x4 16x3 + 61x2 22x 12 = 0, exactly and numerically.
SOLUTION
equation = x4 16 x3 + 61 x2 22 x 12 0; Solve[equation]
{{x 3 5} ,{x 3 + 5} ,{x 5 2 7} ,{x 5 + 2 7}}
0.291503}, {x 0.763932}, {x 5.23607}, {x 10.2915}}
NSolve[equation]
6.4 Solve the following system for w, x, and y and then determine the solution when z = 1, z = 2, and z = 3. w+ x+ y+ z =3 2 w + 3 x + 4 y + 5z = 10 w x + y z =4
SOLUTION
equations = {w + x + y + z 3, 2 w + 3 x + 4 y + 5 z 10,w x + y z 4}; solution = Solve[equations, {w, x, y}]
{{w 1 (5 + 4 z),x 1 z),y 9 z)}} 4 2 4 {{w 9 , x 3 , y 5}} 4 2 4
solution /. z 1
solution /. z 2
{{w 13 , x 5 , y 1}} 4 2 4
solution /. z 3
{{w 17 , x 7 , y 3}} 4 2 4
6.5 Find, to 20 significant digits, a real number such that the sum of itself, its square, and its cube is 30.
SOLUTION
NSolve[x + x2 + x3 30, x, 20]
{{x {x {x
1.8557621138713175532 2.7604410593413850003 }, 1.8557621138713175532 + 2.7604410593413850003 }, 2.7115242277426351064}}
The only real solution is x = 2.7115242277426351064.
6.6 Solve the trigonometric equation 2 sin2 x + 1 = 3 sin x for sin x and then for x.
SOLUTION
To solve for sin x, we can write Solve[2 Sin[x]2 + 1 3 Sin[x]]
{{Sin[x] 1} ,{Sin[x] 1}} 2
Equations
If we solve for x, only the principal solutions (using inverse functions) are obtained. Solve[2 Sin[x]2 + 1 3 Sin[x], x]
Solve ifun : Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
{{x 6 } ,{x 2 }}
Using Reduce we can get all the solutions. Reduce[2 Sin[x]2 + 1 3 Sin[x], x] C[1] Integers & & x = + 2 C[1]||x = + 2 C[1]||x = 5 + 2 C[1] 6 2 6
, , or 5 plus 2 6 6 any integer multiple of 2 x=
6.7 Solve for x : e2x + ex = 3.
SOLUTION
Solve[Exp[2 x] + Exp[x] 3, x]
Solve ifun : Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
x Log 1 ( 1 + 13 ) , x + Log 1 ( 1 13 ) 2 2
Reduce[Exp[2 x] + Exp[x] 3, x] C[1] Integers & &
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