Derive Linear Equations Using Slope - 8th Grade Math
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Find the slope of a line that rises $12$ units while running $-3$ units.
Find the slope of a line that rises $12$ units while running $-3$ units.
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$m=-4$. $m=\frac{12}{-3}=-4$ using rise over run.
$m=-4$. $m=\frac{12}{-3}=-4$ using rise over run.
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Find the slope of the line through $(2,3)$ and $(6,11)$.
Find the slope of the line through $(2,3)$ and $(6,11)$.
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$m=2$. $m=\frac{11-3}{6-2}=\frac{8}{4}=2$
$m=2$. $m=\frac{11-3}{6-2}=\frac{8}{4}=2$
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Find the slope of the line through $(-1,4)$ and $(3,2)$.
Find the slope of the line through $(-1,4)$ and $(3,2)$.
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$m=-\frac{1}{2}$. $m=\frac{2-4}{3-(-1)}=\frac{-2}{4}=-\frac{1}{2}$
$m=-\frac{1}{2}$. $m=\frac{2-4}{3-(-1)}=\frac{-2}{4}=-\frac{1}{2}$
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Find the equation in the form $y=mx$ for a line through the origin and $(5,-10)$.
Find the equation in the form $y=mx$ for a line through the origin and $(5,-10)$.
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$y=-2x$. $m=\frac{-10-0}{5-0}=-2$, so $y=-2x$
$y=-2x$. $m=\frac{-10-0}{5-0}=-2$, so $y=-2x$
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Find the equation of the line with slope $m=-4$ passing through $(0,7)$.
Find the equation of the line with slope $m=-4$ passing through $(0,7)$.
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$y=-4x+7$. Point $(0,7)$ is the y-intercept, so $b=7$.
$y=-4x+7$. Point $(0,7)$ is the y-intercept, so $b=7$.
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What is the equation of a line through the origin with slope $m$?
What is the equation of a line through the origin with slope $m$?
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$y=mx$. No y-intercept term since the line passes through $(0,0)$.
$y=mx$. No y-intercept term since the line passes through $(0,0)$.
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What is slope $m$ defined as in terms of rise and run on a coordinate plane?
What is slope $m$ defined as in terms of rise and run on a coordinate plane?
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$m=\frac{\text{rise}}{\text{run}}$. Vertical change over horizontal change between two points.
$m=\frac{\text{rise}}{\text{run}}$. Vertical change over horizontal change between two points.
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What condition on the points makes the slope formula undefined (a vertical line)?
What condition on the points makes the slope formula undefined (a vertical line)?
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$x_2-x_1=0$. When denominators equal zero, division is undefined.
$x_2-x_1=0$. When denominators equal zero, division is undefined.
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Find $b$ for a line with slope $m=2$ that passes through $(3,1)$ in $y=mx+b$.
Find $b$ for a line with slope $m=2$ that passes through $(3,1)$ in $y=mx+b$.
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$b=-5$. Substitute $(3,1)$: $1=2(3)+b$, so $b=1-6=-5$
$b=-5$. Substitute $(3,1)$: $1=2(3)+b$, so $b=1-6=-5$
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Find the slope $m$ of the line $y=-\frac{3}{5}x+8$.
Find the slope $m$ of the line $y=-\frac{3}{5}x+8$.
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$m=-\frac{3}{5}$. The coefficient of $x$ is the slope in slope-intercept form.
$m=-\frac{3}{5}$. The coefficient of $x$ is the slope in slope-intercept form.
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What point on the coordinate plane represents the $y$-intercept $b$ of $y=mx+b$?
What point on the coordinate plane represents the $y$-intercept $b$ of $y=mx+b$?
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$(0,b)$. The y-intercept occurs where $x=0$.
$(0,b)$. The y-intercept occurs where $x=0$.
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What is the slope-intercept form of a non-vertical line with slope $m$ and $y$-intercept $b$?
What is the slope-intercept form of a non-vertical line with slope $m$ and $y$-intercept $b$?
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$y=mx+b$. Standard form showing slope $m$ and y-intercept $b$.
$y=mx+b$. Standard form showing slope $m$ and y-intercept $b$.
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Find the $y$-intercept $b$ of the line $y=\frac{1}{4}x-6$.
Find the $y$-intercept $b$ of the line $y=\frac{1}{4}x-6$.
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$b=-6$. The constant term is the y-intercept in slope-intercept form.
$b=-6$. The constant term is the y-intercept in slope-intercept form.
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Which property of similar triangles makes slope constant along a non-vertical line?
Which property of similar triangles makes slope constant along a non-vertical line?
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Corresponding side ratios are equal. Similar triangles maintain constant ratios between corresponding sides.
Corresponding side ratios are equal. Similar triangles maintain constant ratios between corresponding sides.
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What ratio of corresponding sides in right triangles on a line equals the slope $m$?
What ratio of corresponding sides in right triangles on a line equals the slope $m$?
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$\frac{\Delta y}{\Delta x}$. Change in y over change in x gives the slope.
$\frac{\Delta y}{\Delta x}$. Change in y over change in x gives the slope.
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Identify the transformation that maps one slope triangle on a line to another: scaling, reflection, or rotation?
Identify the transformation that maps one slope triangle on a line to another: scaling, reflection, or rotation?
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Scaling (dilation). Triangles scale proportionally along a line, maintaining slope.
Scaling (dilation). Triangles scale proportionally along a line, maintaining slope.
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Find the equation of the line through $(2,-1)$ and $(2,5)$ (state the line equation).
Find the equation of the line through $(2,-1)$ and $(2,5)$ (state the line equation).
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$x=2$. Same x-coordinates create a vertical line.
$x=2$. Same x-coordinates create a vertical line.
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What is the slope formula between $ (x_1, y_1) $ and $ (x_2, y_2) $ on a non-vertical line?
What is the slope formula between $ (x_1, y_1) $ and $ (x_2, y_2) $ on a non-vertical line?
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$m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: vertical change divided by horizontal change.
$m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: vertical change divided by horizontal change.
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Find the equation in the form $y=mx+b$ with slope $m=3$ and y-intercept $b=-2$.
Find the equation in the form $y=mx+b$ with slope $m=3$ and y-intercept $b=-2$.
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$y=3x-2$. Substitute given slope and y-intercept into slope-intercept form.
$y=3x-2$. Substitute given slope and y-intercept into slope-intercept form.
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Identify $m$ and $b$ for the line $y=-2x+7$.
Identify $m$ and $b$ for the line $y=-2x+7$.
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$m=-2$, $b=7$. Read coefficient of $x$ for slope, constant for y-intercept.
$m=-2$, $b=7$. Read coefficient of $x$ for slope, constant for y-intercept.
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Find the equation of the line with slope $\frac{1}{2}$ and $y$-intercept $-4$.
Find the equation of the line with slope $\frac{1}{2}$ and $y$-intercept $-4$.
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$y=\frac{1}{2}x-4$. Substitute slope and y-intercept into $y=mx+b$.
$y=\frac{1}{2}x-4$. Substitute slope and y-intercept into $y=mx+b$.
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Find the equation of the line through the origin with slope $-3$.
Find the equation of the line through the origin with slope $-3$.
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$y=-3x$. Use $y=mx$ with $m=-3$ for lines through origin.
$y=-3x$. Use $y=mx$ with $m=-3$ for lines through origin.
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Find the slope between $(-1,4)$ and $(3,0)$.
Find the slope between $(-1,4)$ and $(3,0)$.
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$m=-1$. $m=\frac{0-4}{3-(-1)}=\frac{-4}{4}=-1$
$m=-1$. $m=\frac{0-4}{3-(-1)}=\frac{-4}{4}=-1$
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Identify the geometric fact used to show slope is constant on a line using right triangles.
Identify the geometric fact used to show slope is constant on a line using right triangles.
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Right triangles formed are similar. Similar triangles have proportional sides, ensuring constant slope.
Right triangles formed are similar. Similar triangles have proportional sides, ensuring constant slope.
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What is the $y$-intercept of a line given by $y=mx+b$?
What is the $y$-intercept of a line given by $y=mx+b$?
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$b$. In $y=mx+b$, the constant term is the y-intercept.
$b$. In $y=mx+b$, the constant term is the y-intercept.
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What is the slope of a line given by $y=mx+b$?
What is the slope of a line given by $y=mx+b$?
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$m$. In $y=mx+b$, the coefficient of $x$ is the slope.
$m$. In $y=mx+b$, the coefficient of $x$ is the slope.
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What is the slope-intercept form when the line passes through the origin?
What is the slope-intercept form when the line passes through the origin?
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$y=mx$ (because $b=0$). Origin means $(0,0)$, so y-intercept $b=0$.
$y=mx$ (because $b=0$). Origin means $(0,0)$, so y-intercept $b=0$.
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What point is the $y$-intercept $b$ located at on the coordinate plane?
What point is the $y$-intercept $b$ located at on the coordinate plane?
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$(0,b)$. The y-intercept occurs where the line crosses the y-axis ($x=0$).
$(0,b)$. The y-intercept occurs where the line crosses the y-axis ($x=0$).
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What is the slope-intercept form of a line with slope $m$ and $y$-intercept $b$?
What is the slope-intercept form of a line with slope $m$ and $y$-intercept $b$?
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$y=mx+b$. Standard form showing slope $m$ and y-intercept $b$.
$y=mx+b$. Standard form showing slope $m$ and y-intercept $b$.
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What is the slope of a vertical line (no change in $x$)?
What is the slope of a vertical line (no change in $x$)?
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Undefined. Division by zero (run = 0) makes slope undefined.
Undefined. Division by zero (run = 0) makes slope undefined.
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