Draw the graph for following pair of equations: 5x - 8y + 1 = 0, 3x - (24/5)y + 3/5 = 0

We have the lines 5x - 8y + 1 = 0 and 3x - (24/5)y + 3/5 = 0.

We know that every linear equation represents a straight line, and vice versa. Every point on the line is a solution of the equation. (In general mathematics, we often use the words 'line' and 'equation',and 'point' and 'solution' in the same context. We can call it a solution of the line or a solution of the equation.)  The x and y coordinates of the point determine the value of variables that satisfy the equation. 

A straight line in two dimensional plane (linear equation in two variables) obviously has infinite number of solutions. Substitute any number as value of one variable and obtain the corresponding value of the other variable, and that point will be a solution of the equation.

For drawing any line, we only need two points or solutions. Just to be on the safer side and to avoid any errors, we generally take 3 points. This cross-checks any mistakes. Two easy solutions for any line will be (0, something) and (something, 0). Like, put first variable equal to 0 and obtain the corresponding value of the second, and then put second variable equal to 0 and get the corresponding value of the first. The third solution can be found by substituting anything you like, or anything that would be easy. 


For 5x - 8y + 1 = 0, put x = 0
=>5(0) - 8y + 1 = 0
=> 8y = 1
=> y = 1/8
(0, 1/8) will be a point 

put y = 0
5x - 8(0) + 1 = 0
=> 5x = -1
=> x = -1/5
(-1/5, 0) will be a point

Put x = 3
=> 5(3) - 8y + 1 = 0
=> 15 - 8y + 1 = 0
=>  16 - 8y = 0
=> 8y = 16
=> y = 2
(3, 2) will be a point.

So, points for drawing 5x - 8y + 1 = 0 are (0, 1/8), (-1/5, 0) and (3, 2). Note that these three aren't the only solutions of 5x - 8y + 1 = 0. It has infinite number of solutions, and these are 3 of them, as we need at least any two solutions or points to draw a line.

Follow similar steps for 3x - (24/5)y + 3/5 = 0.
Put x = 0
=> 3(0) - (24/5)y + 3/5 = 0
=> 24y/5 = 3/5
=> 24y = 3
=> y = 1/8
(0, 1/8) will be a solution

Put y = 0
=> 3x - (24/5)0 + 3/5 = 0
=> 3x = -3/5
=> x = -1/5
(-1/5, 0) will be a solution.

Now now now, here is something very interesting. For 5x - 8y + 1 = 0, we found out that (0, 1/8) and (-1/5, 0) satisfy it. While finding solutions for 3x - (24/5)y + 3/5 = 0, we found out that (0, 1/8) and (-1/5, 0) satisfy it as well. Two points, lie on a line, and the same two points lie on another line. This is possible only when the two lines are the same.  

"If any two straight lines (or two linear equations) have two or more than two common points (or common solutions), they are necessarily same lines (or same equations)."

This was easy to guess. One can just see and tell that these two equations are the same. If you multiply both the sides of the second equation by 5/3, you'll get the first equation.

Since both equations are the same, we don't need to find the 3rd point for 3x - (24/5)y + 3/5 = 0. (0, 1/8), (-1/5, 0) and (3, 2) can be used to draw graphs for both the lines, or should we say 'the line', because they are same equations and they are the same lines or one line or coincident lines, whatever suits you best. The graph will look like this -