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

Card 0 of 55

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

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

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

Answer

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

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

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

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

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

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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

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:

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

Compare your answer with the correct one above

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 $\frac{6}{18} =\frac{1}{3}$

Compare your answer with the correct one above

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