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ISEE Middle Level Quantitative : How to find a line on a coordinate plane

Study concepts, example questions & explanations for ISEE Middle Level Quantitative

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Example Questions

Example Question #1 : Geometry

Give the equation of the line through point (1,4) that has slope $-\frac{1}{5}$ .

Possible Answers:

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

y = -5x + 4

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

$y = \frac{1}{5}x - \frac{19}{5}$

y = -5x + 9

Correct answer:

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

Explanation:

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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Example Question #1 : Geometry

Which is the greater quantity?

(A) The slope of the line 2x+y = 15

(B) The slope of the line 3x+y = 15

Possible Answers:

(B) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(A) is greater

Correct answer:

(A) is greater

Explanation:

Rewrite each in the slope-intercept form, y = mx + b ; will be the slope.

2x+y = 15

2x+y -2x = 15-2x

y=15-2x

y = -2x+ 15

The slope of this line is m = -2 .

3x+y = 15

3x+y -3x = 15-3x

y=15-3x

y = -3x+ 15

The slope of this line is m = -3 .

Since -2 > -3 , (A) is greater.

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Example Question #1 : Coordinate Geometry

Which is the greater quantity?

(A) The slope of the line 4x - 2y = 10

(B) The slope of the line y - 2x = 7

Possible Answers:

(B) is greater

(A) is greater

It is impossible to determine which is greater from the information given

(A) and (B) are equal

Correct answer:

(A) and (B) are equal

Explanation:

Rewrite each in the slope-intercept form, y = mx + b ; will be the slope.

4x - 2y = 10

4x - 2y -4x = 10-4x

- 2y = -4x+ 10

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

y = 2x -5

The slope of the line of 4x - 2y = 10 is m = 2

y - 2x = 7

y - 2x + 2x = 7+ 2x

y = 2x + 7

The slope of the line of y - 2x = 7 is also m = 2

The slopes are equal.

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Example Question #4 : How To Find A Line On A Coordinate Plane

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

(a) The slope of the line on the coordinate plane through the points (A+ 1, B+ 1 ) and (A-1, B- 1 ) .

(b) The slope of the line on the coordinate plane through the points (B- 1, A- 1 ) and (B+1, A+ 1 ) .

Possible Answers:

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

Correct answer:

(a) and (b) are equal

Explanation:

The slope of a line through the points (A+ 1, B+ 1 ) and (A-1, B- 1 ) 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 (B- 1, A- 1 ) and (B+1, A+ 1 ) 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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Example Question #5 : How To Find A Line On A Coordinate Plane

A line passes through the points with coordinates (A, B) and (-A, B ) , where A > B >0 . Which expression is equal to the slope of the line?

Possible Answers:

Undefined

$\frac{A}{B}$

$\frac{B}{A}$

Correct answer:

Explanation:

The slope of a line through the points (A, B) and (-A, B ) , 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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ISEE Middle Level Quantitative : How to find a line on a coordinate plane

Study concepts, example questions & explanations for ISEE Middle Level Quantitative

All ISEE Middle Level Quantitative Resources

Example Questions

Example Question #1 : Geometry

Example Question #1 : Geometry

Example Question #1 : Coordinate Geometry

Example Question #4 : How To Find A Line On A Coordinate Plane

Example Question #5 : How To Find A Line On A Coordinate Plane

All ISEE Middle Level Quantitative Resources

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