Rewrite the two equations with variables rather than shapes:.

The first equation can be rewritten in slope-intercept form: 

Its line has slope is , so if the number in the square is known, the slope is known.

is already in slope-intercept form; its line has slope , the number in the circle.

Statement 2 alone gives the number in the circle but provides no clue to the number in the square.

Now assume Statement 1 alone. Then . The slope of the first line is 

,

the slope of the second line. Statement 1 provides sufficient proof that John was successful.

A line with undefined slope is a vertical line.

Infinitely many lines, some vertical and some not, pass through , and infinitely many lines, some vertical and some not, pass through each quadrant, so neither statement alone is sufficient to answer the question.

Now assume both statements are true. Then, since the line passes through Quadrant II, it passes through at least one point with negative -coordinate and positive -coordinate, which we call . Its slope will be

,

a negative slope. Therefore, the slope is not undefined.

Assume both statements are true. The equation takes the form 

,

where  is the number Tim wrote in the triangle.

Put the equation in slope-intercept form:

The coefficient of , which is , is the slope. As can be seen, the value of —that is, the number Tim wrote in the triangle—needs to be known. However, this information is still not given. Whether Tim succeeded is unknown.

The pattern shows a linear equation in slope-intercept form, , with slope  represented by the square and -intercept  represented by the circle. Statement 1 gives the slope, so it provides sufficient information; Statement 2 gives only the -intercept, so it is unhelpful.

The slope of the line of 

can be found by writing this equation in slope-intercept form:

The slope of the line is the coefficient of  is , so Veronica must place the numbers in the shapes to yield an equation whose slope has this equation.

Rewrite the top equation as 

The slope, in terms of  and , can be found similarly:

Its slope is .

Statement 1 asserts that  and  are of unlike sign, so the slope  must be negative. It cannot have sign , so the question is answered.

Assume Statement 2 alone. Then in the above equation, , so the slope is . The slope now depends on the value of , so Statement 2 gives insufficient information.

The -intercept(s) of the graph of  can be found by setting  and solving for :

The graph has exactly one -intercept, .

The -intercept(s) of the graph of  can be found by setting  and solving for :

The graph has exactly one -intercept, .

In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. Each of the two statements yields one of the points, so neither statement alone is sufficient to determine the slope. The two statements together, however, yield two points, and are therefore enough to determine the slope.

The vertex of the parabola of the equation  can be found by first taking , then substituting in the equation and solving for .

The vertex is the point . Since , this is also the -intercept.

In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. From the two statements together, we only know the -intercept and the vertex; however, they are one and the same. Therefore, we have insufficient information to find the slope.

In this case, we are given the midpoint of a line and asked to find one endpoint.

Statement I gives us the other endpoint. We can use this with midpoint formula (see below) to find our other point.

Midpoint formula: 

Statment II gives us the length of the line. However, we know nothing about its orientation or slope. Without some clue as to the steepness of the line, we cannot find the coordinates of its endpoints. You might think we can pull of something with distance formula, but there are going to be two unknowns and one equation, so we are out of luck. 

So,

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Find endpoint Y given the following:

I) Segment RY has its midpoint at (45,65)

II) Point R is located on the x-axis, 13 points from the origin.

I) Gives us the location of the midpoint of our segment

(45,65)

II) Gives us the location of one endpoint

(13,0)

Use I) and II) to work backwards with midpoint formula to find the other endpoint.

So endpoint  is at .

Therefore, both statements are needed to answer the question.

To find the endpoint of a segment, we can generally use the midpoint formula; however, in this case we do not have enough information.

I) Gives us one endpoint

II) Gives us the length of DF

The problem is that we don't know the orientation of DF. It could go in infinitely many directions, so we can't find the location of  without more information.