Coordinate Geometry - ISEE Middle Level Quantitative Reasoning

Card 0 of 11

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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