Find all solutions to the linear system: x+y= 5 2x - y = 3

Solving each equation for y, we have the equivalent system y = 5 - x y = 2x - 3 Setting these expressions for y equal, we have the equation 5 - x = 2x - 3 , which quickly leads to x = 8/3 Substituting for x in the first equation , we have y = 5 - x = 5 - 8/3 = 7/3. Thus the solution is y = 7/3 and x = 8/3

In a right triangle ABC, tan(A) = 3/4. Find sin(A) and cos(A).

Let a be the length of the side opposite angle A, b the length of the side adjacent to angle A and h be the length of the hypotenuse. tan(A) = opposite side / adjacent side = a/b = 3/4 We can say that: a = 3k and b = 4k , where k is a coefficient of proportionality. Let us find h. Pythagora's theorem: h2 = (3k)2 + (5k)2 Solve for h: h = 5k sin(A) = a / h = 3k / 5k = 3/5 and cos(A) = 4k / 5k = 4/5

Find all points of intersections of the circle x2 + 2x + y2 + 4y = -1 and the line x - y = 1

Solve x - y = 1 for x (x = 1 + y) and substitute in the equation of the circle to obtain: (1 + y)2 + 2·(1 + y) + y2 + 4y = -1 Write the above quadratic equation in standard form and solve it to obtain y = - 2 + sqrt(2) and y = - 2 - sqrt(2) Use x = 1 + y to find x Points of intersection: ( -1 + sqrt(2), - 2 + sqrt(2) ) and ( -1 - sqrt(2) , -2 - sqrt(2) )