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To solve it by coordinate graphs you would take a point from the line and plug in the X and Y value into the equations and or inequalities.
You would solve them in exactly the same way as you would solve linear equations with real coefficients. Whether you use substitution or elimination for pairs of equations, or matrix algebra for systems of equations depends on your requirements. But the methods remain the same.
no one really knows how did it came into use as there is no written evidences.
Gaussian elimination is used to solve systems of linear equations.
It does not matter because they are equivalent. You can always convert from a slope-intercept form to a standard linear form (and vice versa).