Coordinate Geometry - ISEE Middle Level Quantitative Reasoning

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Question

Give the equation of the line through point that has slope $-\frac{1}{5}$ .

Answer

Use the point-slope formula with $m = -\frac{1}{5}, x_{1}= 1, y_{1} = 4$

$y -y _{1} = m (x -x _{1} )$

$y -4 = -\frac{1}{5} (x -1 )$

$y -\frac{20}{5} = -\frac{1}{5} x + \frac{1}{5}$

$y = -\frac{1}{5} x + \frac{21}{5}$

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Question

Which is the greater quantity?

(A) The slope of the line

(B) The slope of the line

Answer

Rewrite each in the slope-intercept form, ; will be the slope.

$- 2y \div (-2) =\left ( -4x+ 10 \right )\div (-2)$

The slope of the line of is

The slope of the line of is also

The slopes are equal.

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Question

Which is the greater quantity?

(A) The slope of the line

(B) The slope of the line

Answer

Rewrite each in the slope-intercept form, ; will be the slope.

The slope of this line is .

The slope of this line is .

Since , (A) is greater.

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Question

and are positive integers, and . Which is the greater quantity?

(a) The slope of the line on the coordinate plane through the points and .

(b) The slope of the line on the coordinate plane through the points and .

Answer

The slope of a line through the points and can be found by setting

$x_{1} = A-1,y_{1} = B - 1, x_{2} = A+1, y_{2} = B+ 1$

in the slope formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$= \frac{ (B+ 1)- (B- 1)}{ (A+ 1)- (A- 1)}$

$= \frac{ B-B+ 1+1}{ A-A + 1 + 1}$

$= \frac{ 2}{ 2}$

The slope of a line through the points and can be found similarly:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$= \frac{(A+ 1)- (A- 1) }{ (B+ 1)- (B- 1) }$

$= \frac{A-A + 1+1}{B-B + 1 + 1}$

$= \frac{ 2}{ 2}$

The lines have the same slope.

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Question

A line passes through the points with coordinates and , where . Which expression is equal to the slope of the line?

Answer

The slope of a line through the points and , can be found by setting

$x_{1} = -A,y_{1} =B, x_{2} = A, y_{2} = B$ :

in the slope formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$m = \frac{B-B}{A - (-A)} = \frac{0}{A+A} = \frac{0}{2A} = 0$

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Question

Choose the best answer from the four choices given.

The point (15, 6) is on which of the following lines?

Answer

For this problem, simply plug in the values for the point (15,6) into the different equations (15 for the -value and 6 for the -value) to see which one fits.

$(6)=\frac{1}{2}(15)-7$ (NO)

$(6)=\frac{2}{3}(15)-4$ (YES!)

$(6)=\frac{-1}{2}(15)+7$ (NO)

$(6)=\frac{-2}{3}(15)+4$ (NO)

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Question

Choose the best answer from the four choices given.

What is the point of intersection for the following two lines?

$y=\frac{3}{4}x-11$

$y=\frac{-5}{6}x+8$

Answer

At the intersection point of the two lines the - and - values for each equation will be the same. Thus, we can set the two equations as equal to each other:

$\frac{3}{4}x-11=\frac{-5}{6}x+8$

$\frac{3}{4}x=\frac{-5}{6}x+19$

$\frac{9}{12}x=\frac{-10}{12}x+19$

$\frac{19}{12}x=19$

$(\frac{12}{19})(\frac{19}{12}x)=19 (\frac{12}{19})$

$\large x=12$

$y=\frac{3}{4}(12)-11= -2$

point of intersection $=\left ( 12,-2 \right )$

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Question

Choose the best answer from the four choices given.

What is the -intercept of the line represented by the equation

$y=\frac{7}{9}x-3\ ?$

Answer

In the formula , the y-intercept is represented by (because if you set to zero, you are left with ).

Thus, to find the -intercept, set the value to zero and solve for .

$0=\frac{7}{9}x-3$

$\frac{-7}{9}x=-3$

$(\frac{-9}{7})\frac{-7}{9}x=-3 (\frac{-9}{7})$

$x=\frac{27}{7}=3\frac{6}{7}$

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Question

The ordered pair is in which quadrant?

Answer

There are four quadrants in the coordinate plane. Quadrant I is the top right, and they are numbered counter-clockwise. Since the x-coordinate is , you go to the left one unit (starting from the origin). Since the y-coordinate is , you go upwards four units. Therefore, you are in Quadrant II.

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Question

If angles s and r add up to 180 degrees, which of the following best describes them?

Answer

Two angles that are supplementary add up to 180 degrees. They cannot both be acute, nor can they both be obtuse. Therefore, "Supplementary" is the correct answer.

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Question

The lines of the equations

and

intersect at a point .

Which is the greater quantity?

(a)

(b)

Answer

If and , we can substitute in the second equation as follows:

$4x \div 4 = 8 \div 4$

Substitute:

$y= 3 \cdot 2$

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Question

Give the slope of the line that passes through and .

Answer

Use the slope formula, substituting $x_{1}= 5, y_{1}=3,x_{2}= -1, y_{2}=5$ :

$m =\frac{ y_{2}- y_{1} }{x_{2} - x_{1}}=\frac{ 5-3 }{-1-5} = \frac{ 2 }{-6} =-\frac{1 }{3}$

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Question

What is the slope of the line that passes through and ?

Answer

We can use the slope formula:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{8-4}{8-4}=\frac{4}{4}=1$

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Question

What is the slope of the line that passes through and ?

Answer

We can use the slope formula:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{-7-7}{-3-4}=\frac{-14}{-7}=2$

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Question

Billy set up a ramp for his toy cars. He did this by taking a wooden plank and putting one end on top of a brick that was 3 inches high. He then put the other end on top of a box that was 9 inches high. The bricks were 18 inches apart. What is the slope of the plank?

Answer

The value of the slope (m) is rise over run, and can be calculated with the formula below:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The coordinates of the first end of the plank would be (0,3), given that this is the starting point of the plank (so x would be 0), and y would be 3 since the brick is 3 inches tall.

The coordinates of the second end of the plank would be (18,9) since the plank is 18 inches long (so x would be 18) and y would be 9 since the box was 9 inches tall at the other end.

From this information we know that we can assign the following coordinates for the equation:

$\left(x_{1},y_{1}\right) = (0,3)$ and $(x_{2}, y_{2})= (18,9)$

Below is the solution we would get from plugging this information into the equation for slope:

$m = \frac{9-3}{18-0}}$

This reduces to $\frac{6}{18} =\frac{1}{3}$

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Question

Find the slope of the line that passes through coordinates and .

Answer

The formula for slope is:

$m = \frac{y_{2} - y_{1}}{x_{2}-x_{1}}$

In this particular question our values are given as follows.

$y_{2} = -4$

$y_{1} =-3$

$x_{2} = 8$

$x_{1} = -4$

Substituting the above values into the formula for slope we get,

$\frac{-4 - (-3)}{8- (-4)} = \frac{-1}{12}$

$m = -\frac{1}{12}$ .

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Question

What is the slope of the line depicted by the graph?

Answer

Looking at the graph, it is seen that the line passes through the points (-8,-5) and (8,5).

The slope of a line through the points and can be found by setting

$x_{1} = -8,y_{1} = -5, x_{2} = 8, y_{2} = 5$ :

in the slope formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{5-(-5)}{8 - (-8)} = \frac{5+5}{8 + 8} = \frac{10}{16} = \frac{5}{8}$

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Question

Which of the following is a vertex of the square?

Answer

The coordinates of a point are determined by the distance from the origin. The first point in the ordered pair is the number of units to the left or right of the origin. Negative numbers indicate the number of units to the left while positive numbers indicate the number of units to the right. The second number indicates the number of units above or below the origin. Positive numbers indicate the number of units above while negative numbrs indicate the number of units below the origin. The vertices of the square are:
$\small (-1,2); (-1,4); (1,2); (-1,4)$

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Question

Which of the following points will you find on the -axis?

Answer

A point is located on the -axis if and only if it has -coordinate (first coordinate) 0. Of the five choices, only fits that description.

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Question

Which of the following points is on the -axis?

Answer

A point is located on the -axis if and only if it has a -coordinate equal to zero. So the answer is .

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