how are coordinates (coordinate geometry) useful?

We can co-ordinate geometry to solve a linear equation in 2 variables or even to portray a quadratic polynomial, a linear polynomial etc. This method is very useful, the point when the two lines intersect when drawing a linear equation in 2 variables, tell us the x and y solutions. It is not used as much in maths though, it is used a lot in animation with a z axis to show three-dimensions. It is also a lot in physics, to show the graph of cooling hot water, to show the dependance of current on resistance and potential difference (a straight line tells that resistance is a constant for any value of current and voltage, when material remains the same, length is same, cross-section is same etc) .

Coordinate geometry is supposed to deal mainly with points, mostly in 2D, and is useful to build a basis for points, shapes, lines, functions, etc. Linear Algebra deals with linear (straight) objects which can be used to solve systems of equations, functional analysis, approximations, etc. It helps you conceptualize higher dimensions, as well as give shortcuts and allow you to make approximations easier. My Calc 3 teacher told me that Linear Algebra can make some things, such as Lagrange Multipliers, much easier, by using matrices.