Write as a Vector Equality y=4x-x^2 , y=-3x

Problem

y=4*x−x2,y=−3*x

Solution

1. Set the equations equal to find the intersection points of the two curves by substituting the second equation into the first.

4*x−x2=−3*x

2. Rearrange the equation into standard quadratic form by moving all terms to one side.

x2−7*x=0

3. Factor the quadratic to solve for the values of x

x*(x−7)=0

4. Identify the solutions for x by setting each factor to zero.

x=0

x=7

5. Determine the corresponding y values by substituting the x values back into the linear equation y=−3*x

y=−3*(0)=0

y=−3*(7)=−21

6. Express the intersection points as vectors (v_1) and (v_2) representing the coordinates (x,y)

(v_1)=([0],[0])

(v_2)=([7],[−21])

7. Write the vector equality by defining the vector function r*(x) for both curves and setting them equal to find the points of intersection.

([x],[4*x−x2])=([x],[−3*x])

Final Answer

([x],[4*x−x2])=([x],[−3*x])⇒(x,y)∈{(0,0),(7,−21)}

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