Use Linear Models to Solve Problems - 8th Grade Math
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What are the units of intercept $b$ if $y$ is measured in dollars?
What are the units of intercept $b$ if $y$ is measured in dollars?
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Dollars (same units as $y$). Intercept has the same units as the response variable.
Dollars (same units as $y$). Intercept has the same units as the response variable.
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What does the slope $m$ represent in a linear model $y = mx + b$ for bivariate measurement data?
What does the slope $m$ represent in a linear model $y = mx + b$ for bivariate measurement data?
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Rate of change in $y$ for each $1$-unit increase in $x$. Slope measures how much $y$ changes per unit change in $x$.
Rate of change in $y$ for each $1$-unit increase in $x$. Slope measures how much $y$ changes per unit change in $x$.
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What does the intercept $b$ represent in a linear model $y = mx + b$ for bivariate measurement data?
What does the intercept $b$ represent in a linear model $y = mx + b$ for bivariate measurement data?
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Predicted value of $y$ when $x = 0$. The $y$-intercept is where the line crosses the $y$-axis.
Predicted value of $y$ when $x = 0$. The $y$-intercept is where the line crosses the $y$-axis.
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What is the slope $m$ of the line through $(2,5)$ and $(6,13)$?
What is the slope $m$ of the line through $(2,5)$ and $(6,13)$?
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$m = 2$. Use $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$.
$m = 2$. Use $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$.
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What is the intercept $b$ if $y = 3x + b$ passes through $(4,14)$?
What is the intercept $b$ if $y = 3x + b$ passes through $(4,14)$?
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$b = 2$. Substitute: $14 = 3(4) + b$, so $14 = 12 + b$.
$b = 2$. Substitute: $14 = 3(4) + b$, so $14 = 12 + b$.
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What is the intercept $b$ if slope $m = 2$ and the line passes through $(0,9)$?
What is the intercept $b$ if slope $m = 2$ and the line passes through $(0,9)$?
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$b = 9$. When $x = 0$, $y = b$, so intercept equals $y$-coordinate.
$b = 9$. When $x = 0$, $y = b$, so intercept equals $y$-coordinate.
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Identify the slope and intercept in the model $y = -0.5x + 8$.
Identify the slope and intercept in the model $y = -0.5x + 8$.
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$m = -0.5$, $b = 8$. Coefficient of $x$ is slope; constant term is intercept.
$m = -0.5$, $b = 8$. Coefficient of $x$ is slope; constant term is intercept.
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What is the predicted change in $y$ when $x$ increases by $6$ in the model $y = 1.2x + 5$?
What is the predicted change in $y$ when $x$ increases by $6$ in the model $y = 1.2x + 5$?
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$\Delta y = 7.2$. Multiply slope by change in $x$: $1.2 \times 6 = 7.2$.
$\Delta y = 7.2$. Multiply slope by change in $x$: $1.2 \times 6 = 7.2$.
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What is the linear model equation if slope $m = 4$ and intercept $b = -7$?
What is the linear model equation if slope $m = 4$ and intercept $b = -7$?
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$y = 4x - 7$. Substitute slope and intercept into $y = mx + b$ form.
$y = 4x - 7$. Substitute slope and intercept into $y = mx + b$ form.
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What is the predicted $y$ when $x = 5$ for the model $y = 2x + 3$?
What is the predicted $y$ when $x = 5$ for the model $y = 2x + 3$?
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$y = 13$. Substitute $x = 5$: $y = 2(5) + 3 = 10 + 3$.
$y = 13$. Substitute $x = 5$: $y = 2(5) + 3 = 10 + 3$.
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What is the predicted $x$ when $y = 17$ for the model $y = 2x + 1$?
What is the predicted $x$ when $y = 17$ for the model $y = 2x + 1$?
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$x = 8$. Solve $17 = 2x + 1$: $16 = 2x$, so $x = 8$.
$x = 8$. Solve $17 = 2x + 1$: $16 = 2x$, so $x = 8$.
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What does a slope of $1.5\ \text{cm/hr}$ mean in a model where $x$ is hours and $y$ is plant height?
What does a slope of $1.5\ \text{cm/hr}$ mean in a model where $x$ is hours and $y$ is plant height?
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Each additional $1$ hour predicts $1.5\ \text{cm}$ more height. Slope is the rate of change in context.
Each additional $1$ hour predicts $1.5\ \text{cm}$ more height. Slope is the rate of change in context.
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What does an intercept of $10\ \text{cm}$ mean in a model where $x$ is hours of sunlight and $y$ is height?
What does an intercept of $10\ \text{cm}$ mean in a model where $x$ is hours of sunlight and $y$ is height?
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At $x = 0$, predicted height is $10\ \text{cm}$. Intercept is the starting value when input is zero.
At $x = 0$, predicted height is $10\ \text{cm}$. Intercept is the starting value when input is zero.
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Which variable is the explanatory variable in a model written as $y = mx + b$?
Which variable is the explanatory variable in a model written as $y = mx + b$?
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$x$ is the explanatory (independent) variable. In $y = mx + b$, $x$ is the input variable.
$x$ is the explanatory (independent) variable. In $y = mx + b$, $x$ is the input variable.
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Which variable is the response variable in a model written as $y = mx + b$?
Which variable is the response variable in a model written as $y = mx + b$?
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$y$ is the response (dependent) variable. In $y = mx + b$, $y$ depends on $x$.
$y$ is the response (dependent) variable. In $y = mx + b$, $y$ depends on $x$.
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What are the units of slope if $x$ is in hours and $y$ is in centimeters?
What are the units of slope if $x$ is in hours and $y$ is in centimeters?
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$\text{cm/hr}$. Slope units are output units per input unit.
$\text{cm/hr}$. Slope units are output units per input unit.
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What is the slope $m$ if $y$ increases by $12$ when $x$ increases by $3$?
What is the slope $m$ if $y$ increases by $12$ when $x$ increases by $3$?
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$m = 4$. Slope equals change in $y$ divided by change in $x$: $\frac{12}{3}$.
$m = 4$. Slope equals change in $y$ divided by change in $x$: $\frac{12}{3}$.
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What is the meaning of a negative slope in a bivariate linear model?
What is the meaning of a negative slope in a bivariate linear model?
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As $x$ increases, predicted $y$ decreases. Negative slope means inverse relationship.
As $x$ increases, predicted $y$ decreases. Negative slope means inverse relationship.
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What is the meaning of a slope of $0$ in a bivariate linear model?
What is the meaning of a slope of $0$ in a bivariate linear model?
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Predicted $y$ stays constant as $x$ changes. Zero slope means a horizontal line.
Predicted $y$ stays constant as $x$ changes. Zero slope means a horizontal line.
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What is the slope formula using two data points $ (x_1, y_1) $ and $ (x_2, y_2) $?
What is the slope formula using two data points $ (x_1, y_1) $ and $ (x_2, y_2) $?
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$m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run formula calculates slope between two points.
$m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run formula calculates slope between two points.
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Find the slope of the line through $ (2,7) $ and $ (6,15) $.
Find the slope of the line through $ (2,7) $ and $ (6,15) $.
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$m = 2$. Use $m = \frac{15-7}{6-2} = \frac{8}{4}$.
$m = 2$. Use $m = \frac{15-7}{6-2} = \frac{8}{4}$.
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Which value is the $y$-intercept in the model $y = 0.75x - 6$?
Which value is the $y$-intercept in the model $y = 0.75x - 6$?
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$-6$. Y-intercept is the constant term in $y = mx + b$.
$-6$. Y-intercept is the constant term in $y = mx + b$.
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Identify the meaning of $b$ in $y = 1.5x + 10$ if $x$ is hours of sunlight and $y$ is plant height (cm).
Identify the meaning of $b$ in $y = 1.5x + 10$ if $x$ is hours of sunlight and $y$ is plant height (cm).
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At $x = 0$, predicted height is $10$ cm. Y-intercept occurs when input is zero.
At $x = 0$, predicted height is $10$ cm. Y-intercept occurs when input is zero.
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What does the phrase "predicted value" mean in a linear model context?
What does the phrase "predicted value" mean in a linear model context?
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The model’s estimate of $y$ for a given $x$. Linear models predict output from input values.
The model’s estimate of $y$ for a given $x$. Linear models predict output from input values.
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What does a slope of $0$ mean about the relationship between $x$ and $y$ in a linear model?
What does a slope of $0$ mean about the relationship between $x$ and $y$ in a linear model?
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Predicted $y$ does not change as $x$ changes. Zero slope means horizontal line.
Predicted $y$ does not change as $x$ changes. Zero slope means horizontal line.
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What does a negative slope mean about the association between $x$ and $y$ in a linear model?
What does a negative slope mean about the association between $x$ and $y$ in a linear model?
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As $x$ increases, predicted $y$ decreases. Negative slope means inverse relationship.
As $x$ increases, predicted $y$ decreases. Negative slope means inverse relationship.
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What is the $y$-intercept in a linear model $y = mx + b$ and what does it represent in context?
What is the $y$-intercept in a linear model $y = mx + b$ and what does it represent in context?
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$b$; the predicted value of $y$ when $x = 0$. Y-intercept is where the line crosses the y-axis.
$b$; the predicted value of $y$ when $x = 0$. Y-intercept is where the line crosses the y-axis.
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What is the slope in a linear model $y = mx + b$ and what does it represent in context?
What is the slope in a linear model $y = mx + b$ and what does it represent in context?
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$m$; the change in $y$ for each $1$-unit increase in $x$. Slope measures rate of change between variables.
$m$; the change in $y$ for each $1$-unit increase in $x$. Slope measures rate of change between variables.
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A model is $y = -1.2x + 50$. What is the predicted change in $y$ when $x$ increases by $5$?
A model is $y = -1.2x + 50$. What is the predicted change in $y$ when $x$ increases by $5$?
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Decrease of $6$. Change is slope times change in $x$: $-1.2 imes 5$.
Decrease of $6$. Change is slope times change in $x$: $-1.2 imes 5$.
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A model is $y = 4x + 20$. What is the predicted change in $y$ when $x$ increases by $3$?
A model is $y = 4x + 20$. What is the predicted change in $y$ when $x$ increases by $3$?
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Increase of $12$. Change is slope times change in $x$: $4 imes 3$.
Increase of $12$. Change is slope times change in $x$: $4 imes 3$.
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