The graph shows one of the linear equations for a system of equations. which equation represents the second linear equation for the system of equations that has the solution (3/4, 5\6)?
a) 2x - 3y = 4
b) 2x + 3y = -4
c) 2/3x + 3y = 3
d) 2/3x - 3y = 3

  C)  2/3x + 3y = 3

Step-by-step explanation:

The attached graph shows only choice C intersects the desired point.

Essentially, the given graph is irrelevant. What you want to know is which equation has (3/4, 5/6) as a solution.

There are at least a couple of ways you can figure this out:

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If we consider the second approach, we realize all of the equations have negative y-intercepts except choice C*. For an equation with a negative y-intercept to be the solution, the x-intercept must be less than 3/4. That is not the case for any of choices A, B, or D, leaving only choice C as a viable possibility.

We can check to see if the given point satisfies choice C. Filling in for x and y, we get ...

  (2/3)(3/4) + 3(5/6) = 2/4 +5/2 = 3 . . . . . . the point is on this line

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In the attached graph, choice C is the black line.

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* For an equation of the form ax+by=c, the y-intercept is c/b and the x-intercept is c/a. The sign of the quotient can be figured from the signs of a, b, and c without doing any division.

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Comment on intercepts

Considering the signs and values of the x- and y-intercepts can tell you a lot about the solution to a system of equations. They tell you where the line segment between the axes lies, and they can give you a clue as to the location (quadrant) of the intersection point of two lines. Often, the intercepts are all you need to create a useful graph of an equation.