Question 1105330
if it has no solutions, the lines are parallel.


if it has infinitely many solutions, the lines are identical.


if it has one solution, then the lines are neither parallel nor identical.


how do you determine?


the lines are identical if the the equations have the same slope and the wsame y-intercept.


the lines are parallel if the equations have the same slope but the y-intercepts are different.


the lines are neither parallel nor identical if the lines do not have the same slope.


your equation is in standard form of ax + by =c


to find the slope and the y-intercept, convert the equation to slope intercept form of y = mx + b, where m is the slope and b is the y-intercept.


that is done as shown below:


start with 7x - 3y = 2
add 3y to both sides of the equation and subtract 2 from both sides of the equation to get 7x - 2 = 3y
divide both sides of the equation by 3 to get 7/3 * x - 2/3 = y
flip sides of the equation to get y = 7/3 * x - 2/3


y = 7/3 * x - 2/3 is the slope intercept form of the equation 7x - 3y = 2


m = 7/3
b = -2/3


a line parallel to the line created by this equation will have the same slope but a different y-intercept.


any y-intercept will do as long as it's different.


i chose a y-intercept of 7.


that makes the slope intercept form of the equation equal to y = 7/3 * x - 7


convert that equation to standard form by doing something like the following:


add 7 to both sides of the equation and subtract y from both sides of the equation to get 7 = 7/3 * x - y
flip sides to get 7/3 * x - y = 7
multiply both sides of the equation by 3 to get 7x - 3y = 21


7x - 3y = 21 is the standard form of the equation y = 7/3 * x - 7.


to find an equation that will give you infinitely many solutions to the original equation of 7x - 3y = 2, simply find an equation that is a multiple of it.


for example, the equation of 21x - 9y = 6 would be such an equation.


when you solve for the slope intercept form of this equation, it should be identical to the slope intercept form of the original equation.


add 9y to both sides of the equation and subtract 6 from both sides of the equation to get 21x - 6 = 9y
divide both sides of the equation by 9 to get 21/9 * x - 6/9 = y
simplify to get 7/3 * x - 2/3 = y
flip sides to get y = 7/3 * x - 2/3


this is identical to the slope intercept form of the original equation and therefore will have infinitely many solutions.


to summarize:


original equation is 7x - 3y = 2
parallel equation is 7x - 3y = 7
identical equation is 21x - 9y = 6


their slope intercept forms are:


original equation is y = -7/3 * x - 2/3
parallel equation is y = -7/3 * x - 7/3
identical equation is y = -7/3 * x - 2/3


the following graphs will show this to be true.


the first graph uses the standard form of the equations.
the second graph uses the slope intercept form of the equation.
the third graph combines them to show you that the different forms of the equations create the same graph.


the two blue equations create the same blue line meaning that the blue equations are identical.
the red equation creates the red line that is clearly parallel to the blue line.


in the last graph, all the blue equations create the same line and all the red equations create the same parallel line, confirming that the standard form of their corresponding equations and the slope intercept form of their corresponding equations are identical.


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