The graphed line shown below is y=-3x+6

From the given graph, we can know two points:

Using the slope intercept form of the equation of a line, we get:

Where the slope is and the y-intercept is

From mathematics, we know that a system of two linear equations has no solution if and only if the two equations have the same slope, but not the same y-intercept. So, from the options, the second equation is:

And the slope intercept form of this equation is:

So the slope is and the y-intercept is , the systems formed by these two equations has no solution.

The equation, when graphed together with the line , which will form a system of equations with no solution is , meaning the second option on the picture.

Systems of equations are systems of n equations with n unknowns (in this case, n would be 2 since we only have 2 equations and 2 unknowns, x and y). In the most general case, having 2 equations with 2 unknowns will able us to solve uniquely for those unknowns, however there are some cases in which this is not possible (our answer lies in those cases).

There are many ways to solve this problem (all equivalent), in this case we will use a graphical interpretation since it's the most simple. In the picture, there is a drawn line whose equation is , if we draw another line on the cartesian plane then both lines will eventualy intersect (meaning a system of equations with those 2 equations will have a unique solution)... unless both lines are parallel.

In mathematical terms, parallel lines share the same slope (you can find more information about that in the attached link if necessary). It is very easy to find the slope of a line from its equation, since lines of the form will have slope m.

Therefor, the solution to our problem will be to find a parallel line to that drawn in the picture, which has slope equal to -3. From the given options, only the second and third options share this same slope.

Even though the third option, meaning has slope -3, it is equivalent to the drawn line (they are exactly the same equation, so both lines intersect in every point). Because of this, a system of equations formed from these 2 equations can have a set of solutions (this is a very special case, I recommend you to check for the Rouche-Capelli theorem).

Therefor, the equation which guarantees to have a system of equations with no solution is the second option, since they are both different parallel lines.

System of equations, intersection, parallel, consistency

360 different combinations

linear

step-by-step explanation: