Consider the two equations, y=5(x-2)^2-6 and (y+2)^2=9
how many solutions are there to this system? without solving, give a convincing argument explain why there are that many solutions to the system.

One solution. A system of linear equations has one solution when the graphs intersect at a point.

1\small{1}

1

2\small{2}

2

3\small{3}

3

-1\small{\llap{-}1}

-

1

1\small{1}

1

2\small{2}

2

3\small{3}

3

-1\small{\llap{-}1}

-

1

yy

y

xx

x

No solution. A system of linear equations has no solution when the graphs are parallel.

1\small{1}

1

2\small{2}

2

3\small{3}

3

-1\small{\llap{-}1}

-

1

1\small{1}

1

2\small{2}

2

3\small{3}

3

-1\small{\llap{-}1}

-

1

yy

y

xx

x

Infinite solutions. A system of linear equations has infinite solutions when the graphs are the exact same line.

Step-by-step explanation:

x coordinate can be written by using the formula, x= x1 + 2/5(x2 - x1)

step-by-step explanation:

formula

x = x1 + p/w(x2 - x1)

let the line segment ab with a(x1,y1) and b(x2,y2)

a point(x,y) divide ab in the ratio, 2: 3 the x coordinate can be written as

x = x1 + p/w(x2 - x1)

here p=2 and w = 2+3

therefore,

x coordinate can be written by using the formula, x= x1 + 2/5(x2 - x1)

similarly y coordinate y = y1 + 2/5(y2 - y1)