Finding Slope - ISEE Upper Level: Mathematics Achievement
Card 1 of 25
Which condition makes the slope equal to $0$ when using $m=\frac{y_2-y_1}{x_2-x_1}$?
Which condition makes the slope equal to $0$ when using $m=\frac{y_2-y_1}{x_2-x_1}$?
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$y_2-y_1=0$. A zero numerator indicates no vertical change, yielding a horizontal line with zero slope.
$y_2-y_1=0$. A zero numerator indicates no vertical change, yielding a horizontal line with zero slope.
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What is the slope of the line through $(2,3)$ and $(6,11)$?
What is the slope of the line through $(2,3)$ and $(6,11)$?
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$m=2$. Apply the slope formula: $(11-3)/(6-2)=8/4=2$, representing rise over run.
$m=2$. Apply the slope formula: $(11-3)/(6-2)=8/4=2$, representing rise over run.
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Which condition makes the slope undefined when using $m=\frac{y_2-y_1}{x_2-x_1}$?
Which condition makes the slope undefined when using $m=\frac{y_2-y_1}{x_2-x_1}$?
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$x_2-x_1=0$. A zero denominator indicates a vertical line, where slope cannot be defined.
$x_2-x_1=0$. A zero denominator indicates a vertical line, where slope cannot be defined.
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State the slope formula for the line through points $(x_1,y_1)$ and $(x_2,y_2)$.
State the slope formula for the line through points $(x_1,y_1)$ and $(x_2,y_2)$.
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$m=\frac{y_2-y_1}{x_2-x_1}$. The slope measures the rate of change as the vertical difference divided by the horizontal difference between two points.
$m=\frac{y_2-y_1}{x_2-x_1}$. The slope measures the rate of change as the vertical difference divided by the horizontal difference between two points.
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What is the slope of the line through $(-1,4)$ and $(3,-2)$?
What is the slope of the line through $(-1,4)$ and $(3,-2)$?
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$m=-\frac{3}{2}$. Using the formula, $(-2-4)/(3-(-1))=-6/4=-3/2$, indicating a negative slope.
$m=-\frac{3}{2}$. Using the formula, $(-2-4)/(3-(-1))=-6/4=-3/2$, indicating a negative slope.
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What is the slope of the line through $(0,0)$ and $(5,-10)$?
What is the slope of the line through $(0,0)$ and $(5,-10)$?
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$m=-2$. Calculate $(-10-0)/(5-0)=-10/5=-2$, showing a downward slope.
$m=-2$. Calculate $(-10-0)/(5-0)=-10/5=-2$, showing a downward slope.
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What is the slope of the line through $(7,2)$ and $(7,9)$?
What is the slope of the line through $(7,2)$ and $(7,9)$?
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Slope is undefined. Identical x-coordinates result in division by zero, making the slope undefined for vertical lines.
Slope is undefined. Identical x-coordinates result in division by zero, making the slope undefined for vertical lines.
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What is the slope of the line through $(-3,5)$ and $(4,5)$?
What is the slope of the line through $(-3,5)$ and $(4,5)$?
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$m=0$. Identical y-coordinates yield a numerator of zero, resulting in a slope of zero for horizontal lines.
$m=0$. Identical y-coordinates yield a numerator of zero, resulting in a slope of zero for horizontal lines.
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Identify the slope type for a horizontal line (in terms of $m$).
Identify the slope type for a horizontal line (in terms of $m$).
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$m=0$. Horizontal lines have no change in y, so the slope is zero regardless of x-change.
$m=0$. Horizontal lines have no change in y, so the slope is zero regardless of x-change.
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Identify the slope type for a vertical line (in terms of $m$).
Identify the slope type for a vertical line (in terms of $m$).
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Slope is undefined. Vertical lines have no change in x, causing division by zero and an undefined slope.
Slope is undefined. Vertical lines have no change in x, causing division by zero and an undefined slope.
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What is the slope of the line through $(-6,1)$ and $(0,-2)$?
What is the slope of the line through $(-6,1)$ and $(0,-2)$?
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$m=-\frac{1}{2}$. Compute $(-2-1)/(0-(-6))=-3/6=-1/2$.
$m=-\frac{1}{2}$. Compute $(-2-1)/(0-(-6))=-3/6=-1/2$.
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What is the slope of the line through $(2,-5)$ and $(10,-1)$?
What is the slope of the line through $(2,-5)$ and $(10,-1)$?
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$m=\frac{1}{2}$. Formula results in $(-1-(-5))/(10-2)=4/8=1/2$.
$m=\frac{1}{2}$. Formula results in $(-1-(-5))/(10-2)=4/8=1/2$.
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What is the slope of the line through $(4,7)$ and $(0,1)$?
What is the slope of the line through $(4,7)$ and $(0,1)$?
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$m=\frac{3}{2}$. Calculate $(1-7)/(0-4)=-6/(-4)=3/2$.
$m=\frac{3}{2}$. Calculate $(1-7)/(0-4)=-6/(-4)=3/2$.
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Find and correct the error: $m=\frac{x_2-x_1}{y_2-y_1}$ for slope between two points.
Find and correct the error: $m=\frac{x_2-x_1}{y_2-y_1}$ for slope between two points.
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Correct: $m=\frac{y_2-y_1}{x_2-x_1}$. The incorrect formula inverts rise over run; the correct one uses change in y over change in x.
Correct: $m=\frac{y_2-y_1}{x_2-x_1}$. The incorrect formula inverts rise over run; the correct one uses change in y over change in x.
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What happens to the slope value if you swap the two points in $m=\frac{y_2-y_1}{x_2-x_1}$?
What happens to the slope value if you swap the two points in $m=\frac{y_2-y_1}{x_2-x_1}$?
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Slope stays the same. Swapping points negates both numerator and denominator, preserving the slope value.
Slope stays the same. Swapping points negates both numerator and denominator, preserving the slope value.
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What is the slope of the line through $(-2,9)$ and $(-8,3)$?
What is the slope of the line through $(-2,9)$ and $(-8,3)$?
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$m=1$. Compute $(3-9)/(-8-(-2))=-6/(-6)=1$.
$m=1$. Compute $(3-9)/(-8-(-2))=-6/(-6)=1$.
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What is the slope of the line through $(-2,-1)$ and $(2,7)$?
What is the slope of the line through $(-2,-1)$ and $(2,7)$?
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$m=2$. Using $(7-(-1))/(2-(-2))=8/4=2$, the slope is positive.
$m=2$. Using $(7-(-1))/(2-(-2))=8/4=2$, the slope is positive.
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What is the slope of the line through $(1,-2)$ and $(4,4)$?
What is the slope of the line through $(1,-2)$ and $(4,4)$?
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$m=2$. Compute $(4-(-2))/(4-1)=6/3=2$ using the slope formula.
$m=2$. Compute $(4-(-2))/(4-1)=6/3=2$ using the slope formula.
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What is the slope of the line through $(-5,2)$ and $(-1,10)$?
What is the slope of the line through $(-5,2)$ and $(-1,10)$?
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$m=2$. Formula gives $(10-2)/(-1-(-5))=8/4=2$.
$m=2$. Formula gives $(10-2)/(-1-(-5))=8/4=2$.
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What is the slope of the line through $(8,-3)$ and $(2,9)$?
What is the slope of the line through $(8,-3)$ and $(2,9)$?
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$m=-2$. Calculate $(9-(-3))/(2-8)=12/(-6)=-2$, a negative value.
$m=-2$. Calculate $(9-(-3))/(2-8)=12/(-6)=-2$, a negative value.
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What is the slope of the line through $(3,1)$ and $(9,0)$?
What is the slope of the line through $(3,1)$ and $(9,0)$?
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$m=-\frac{1}{6}$. Slope is $(0-1)/(9-3)=-1/6$, showing a slight decline.
$m=-\frac{1}{6}$. Slope is $(0-1)/(9-3)=-1/6$, showing a slight decline.
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What is the slope of the line through $(-4,-6)$ and $(2,3)$?
What is the slope of the line through $(-4,-6)$ and $(2,3)$?
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$m=\frac{3}{2}$. Using $(3-(-6))/(2-(-4))=9/6=3/2$.
$m=\frac{3}{2}$. Using $(3-(-6))/(2-(-4))=9/6=3/2$.
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What is the slope of the line through $(5,12)$ and $(1,4)$?
What is the slope of the line through $(5,12)$ and $(1,4)$?
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$m=2$. Order of points yields $(4-12)/(1-5)=-8/(-4)=2$.
$m=2$. Order of points yields $(4-12)/(1-5)=-8/(-4)=2$.
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What is the slope of the line through $(2,5)$ and $(-1,-4)$?
What is the slope of the line through $(2,5)$ and $(-1,-4)$?
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$m=3$. Using $(-4-5)/(-1-2)=-9/(-3)=3$.
$m=3$. Using $(-4-5)/(-1-2)=-9/(-3)=3$.
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What is the slope of the line through $(-3,8)$ and $(1,0)$?
What is the slope of the line through $(-3,8)$ and $(1,0)$?
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$m=-2$. Slope is $(0-8)/(1-(-3))=-8/4=-2$.
$m=-2$. Slope is $(0-8)/(1-(-3))=-8/4=-2$.
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