Write as a Vector Equality y=5x-5 , y=x-3

Problem

y=5*x−5,y=x−3

Solution

1. Set the equations equal to find the intersection point of the two lines by substituting the first expression for y into the second equation.

5*x−5=x−3

2. Solve for x by subtracting x from both sides and adding 5 to both sides.

4*x=2

x=1/2

3. Solve for y by substituting the x value back into one of the original linear equations.

y=1/2−3

y=−5/2

4. Identify the intersection point as the position vector (r_0) and the slopes of the lines to determine their direction vectors.

(r_0)=([1/2],[−5/2])

5. Express as a vector equality by representing the system as a vector equation where the vector ([x],[y]) satisfies both conditions, or more commonly, by representing the intersection as a single vector coordinate.

([x],[y])=([1/2],[−5/2])

Final Answer

([x],[y])=([1/2],[−5/2])

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