How to find the points on a coordinate plane - ISEE Middle Level Quantitative Reasoning

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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

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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Question

A line segment on the coordinate plane has its endpoints at the points with coordinates $\left ( 3 \frac{1}{3}, - 1 \frac{1}{2}\right )$ and $\left ( 10, 5 \frac{1}{4}\right )$ . Give the coordinates of the midpoint of the segment.

Answer

The -coordinate of the midpoint is half the sum of the -coordinates of the endpoints:

$\frac{1}{2} \times \left ( 3 \frac{1}{3} + 10 \right )$

$= \frac{1}{2} \times 13 \frac{1}{3}$

$= \frac{1}{2} \times \frac{40}{3}$

$= \frac{1}{1} \times \frac{20}{3}$

$= \frac{20}{3}$

$= 6 \frac{1}{3}$

The -coordinate of the midpoint is found similarly:

$\frac{1}{2} \times \left ( - 1 \frac{1}{2} + 5 \frac{1}{4} \right )$

$= \frac{1}{2} \times \left ( 5 \frac{1}{4} - 1 \frac{1}{2} \right )$

$= \frac{1}{2} \times \left ( 5 \frac{1}{4} - 1 \frac{2}{4} \right )$

$= \frac{1}{2} \times \left ( 4 \frac{5}{4} - 1 \frac{2}{4} \right )$

$= \frac{1}{2} \times 3 \frac{3}{4}$

$= \frac{1}{2} \times \frac{15}{4}$

$= \frac{ 15}{8}$

$=1 \frac{7}{8}$

The correct response is $\left (6 \frac{2}{3}, 1 \frac{7}{8}\right )$ .

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Question

A line segment on the coordinate plane has its endpoints at the points with coordinates $\left ( - 300, 200 \right )$ and $\left ( -200, 300 \right )$ . Give the coordinates of the midpoint of the segment.

Answer

The -coordinate of the midpoint can be found by dividing the sum of the -coordinates of the endpoints by 2:

$[ -300 + (-200) ] \div 2 = [ -(300 + 200) ] \div 2 = -500 \div 2 = - (500 \div 2) = -250$

The -coordinate of the midpoint is found similarly:

$(200 + 300) \div 2 = 500 \div 2 = 250$ .

The correct response is .

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Question

A line segment on the coordinate plane has its endpoints at the points with coordinates and . Give the midpoint of the segment.

Answer

The -coordinate of the midpoint can be found by dividing the sum of the -coordinates of the endpoints by 2:

$(7.32 + 10) \div 2 = 17.32 \div 2 = 8.66$

The -coordinate of the midpoints is found similarly:

$(-3.16 + 10) \div 2 =(10 -3.16 ) \div 2 = 6.84 \div 2 = 3.42$

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Question

Angela plotted points A through E on the coordinate plane shown below.

Which point is located at the ordered pair

Answer

Start at point (0, 0) in the bottom, left corner of the the coordinate plane. From there, count right 2 spaces. This is the first number in the given ordered pair. From that point, count up 8 spaces. This is the second number in the given ordered pair.

By moving right 2 and up 8, you should have found Point B.

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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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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:

$y= 3 \cdot 2$

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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 .

Compare your answer with the correct one above

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