Question A hiker hikes at a steady rate on a mountain. Which student wrote the correct equation $y = -125x + 5,775$ to represent the linear relationship between hours hiked $(x)$ and altitude $(y)$ ?

Studdy Solution

STEP 1

Assumptions
1. The relationship between the number of hours hiked ( $x$ ) and the current altitude ( $y$ ) is linear.
2. We have a table with values of $x$ and corresponding values of $y$ .
3. We need to determine which student's equation correctly represents the linear relationship.

STEP 2

To find the correct equation, we first need to determine the slope of the line. The slope ( $m$ ) is the change in altitude per hour hiked, which can be calculated using two points from the table.
$m = \frac{\Delta y}{\Delta x}$

STEP 3

Choose two points from the table to calculate the slope. For example, we can use the points (1, 5650) and (2, 5525).
$m = \frac{y_2 - y_1}{x_2 - x_1}$

STEP 4

Substitute the chosen points into the slope formula.
$m = \frac{5525 - 5650}{2 - 1}$

STEP 5

Calculate the slope.
$m = \frac{-125}{1} = -125$

STEP 6

Now that we have the slope, we need to find the y-intercept ( $b$ ). The y-intercept is the altitude when $x$ (the number of hours hiked) is 0.

STEP 7

To find the y-intercept, we can use the slope and one of the points from the table to plug into the point-slope form of the linear equation $y - y_1 = m(x - x_1)$ .

STEP 8

Choose a point from the table. For simplicity, let's use the first point (1, 5650).
$y - 5650 = -125(x - 1)$

STEP 9

Now, solve for $y$ to find the y-intercept.
$y = -125(x - 1) + 5650$

STEP 10

Distribute the slope through the parentheses.
$y = -125x + 125 + 5650$

STEP 11

Combine like terms to find the y-intercept.
$y = -125x + 5775$

STEP 12

Compare the equation we found with the students' equations.
1. Mateo's equation: $y = 125x + 5775$
2. Julie's equation: $y = -125x + 5775$
3. Oliver's equation: $y = -125x + 5650$

STEP 13

Identify the correct equation by matching the slope and y-intercept we calculated.
The correct equation is Julie's equation: $y = -125x + 5775$ .

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Question A hiker hikes at a steady rate on a mountain. Which student wrote the correct equation $y = -125x + 5,775$ to represent the linear relationship between hours hiked $(x)$ and altitude $(y)$ ?

Studdy Solution

STEP 1

Assumptions
1. The relationship between the number of hours hiked ( $x$ ) and the current altitude ( $y$ ) is linear.
2. We have a table with values of $x$ and corresponding values of $y$ .
3. We need to determine which student's equation correctly represents the linear relationship.

STEP 2

To find the correct equation, we first need to determine the slope of the line. The slope ( $m$ ) is the change in altitude per hour hiked, which can be calculated using two points from the table.
$m = \frac{\Delta y}{\Delta x}$

STEP 3

Choose two points from the table to calculate the slope. For example, we can use the points (1, 5650) and (2, 5525).
$m = \frac{y_2 - y_1}{x_2 - x_1}$

STEP 4

Substitute the chosen points into the slope formula.
$m = \frac{5525 - 5650}{2 - 1}$

STEP 5

Calculate the slope.
$m = \frac{-125}{1} = -125$

STEP 6

Now that we have the slope, we need to find the y-intercept ( $b$ ). The y-intercept is the altitude when $x$ (the number of hours hiked) is 0.

STEP 7

To find the y-intercept, we can use the slope and one of the points from the table to plug into the point-slope form of the linear equation $y - y_1 = m(x - x_1)$ .

STEP 8

Choose a point from the table. For simplicity, let's use the first point (1, 5650).
$y - 5650 = -125(x - 1)$

STEP 9

Now, solve for $y$ to find the y-intercept.
$y = -125(x - 1) + 5650$

STEP 10

Distribute the slope through the parentheses.
$y = -125x + 125 + 5650$

STEP 11

Combine like terms to find the y-intercept.
$y = -125x + 5775$

STEP 12

Compare the equation we found with the students' equations.
1. Mateo's equation: $y = 125x + 5775$
2. Julie's equation: $y = -125x + 5775$
3. Oliver's equation: $y = -125x + 5650$

STEP 13

Identify the correct equation by matching the slope and y-intercept we calculated.
The correct equation is Julie's equation: $y = -125x + 5775$ .

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Question A hiker hikes at a steady rate on a mountain. Which student wrote the correct equation y=−125x+5,775y = -125x + 5,775y=−125x+5,775 to represent the linear relationship between hours hiked (x)(x)(x) and altitude (y)(y)(y)?

Studdy Solution

STEP 1

Assumptions1. The relationship between the number of hours hiked (xxx) and the current altitude (yyy) is linear.2. We have a table with values of xxx and corresponding values of yyy.3. We need to determine which student's equation correctly represents the linear relationship.

STEP 2

To find the correct equation, we first need to determine the slope of the line. The slope (mmm) is the change in altitude per hour hiked, which can be calculated using two points from the table.m=ΔyΔxm = \frac{\Delta y}{\Delta x}m=ΔxΔy​

STEP 3

Choose two points from the table to calculate the slope. For example, we can use the points (1, 5650) and (2, 5525).m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}m=x2​−x1​y2​−y1​​

STEP 4

Substitute the chosen points into the slope formula.m=5525−56502−1m = \frac{5525 - 5650}{2 - 1}m=2−15525−5650​

STEP 5

Calculate the slope.m=−1251=−125m = \frac{-125}{1} = -125m=1−125​=−125

STEP 6

Now that we have the slope, we need to find the y-intercept (bbb). The y-intercept is the altitude when xxx (the number of hours hiked) is 0.

STEP 7

To find the y-intercept, we can use the slope and one of the points from the table to plug into the point-slope form of the linear equation y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​).

STEP 8

Choose a point from the table. For simplicity, let's use the first point (1, 5650).y−5650=−125(x−1)y - 5650 = -125(x - 1)y−5650=−125(x−1)

STEP 9

Now, solve for yyy to find the y-intercept.y=−125(x−1)+5650y = -125(x - 1) + 5650y=−125(x−1)+5650

STEP 10

Distribute the slope through the parentheses.y=−125x+125+5650y = -125x + 125 + 5650y=−125x+125+5650

STEP 11

Combine like terms to find the y-intercept.y=−125x+5775y = -125x + 5775y=−125x+5775

STEP 12

Compare the equation we found with the students' equations.1. Mateo's equation: y=125x+5775y = 125x + 5775y=125x+57752. Julie's equation: y=−125x+5775y = -125x + 5775y=−125x+57753. Oliver's equation: y=−125x+5650y = -125x + 5650y=−125x+5650

STEP 13

Identify the correct equation by matching the slope and y-intercept we calculated.The correct equation is Julie's equation: y=−125x+5775y = -125x + 5775y=−125x+5775.

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Question A hiker hikes at a steady rate on a mountain. Which student wrote the correct equation $y = -125x + 5,775$ to represent the linear relationship between hours hiked $(x)$ and altitude $(y)$ ?

Assumptions
1. The relationship between the number of hours hiked ( $x$ ) and the current altitude ( $y$ ) is linear.
2. We have a table with values of $x$ and corresponding values of $y$ .
3. We need to determine which student's equation correctly represents the linear relationship.

To find the correct equation, we first need to determine the slope of the line. The slope ( $m$ ) is the change in altitude per hour hiked, which can be calculated using two points from the table.
$m = \frac{\Delta y}{\Delta x}$

Choose two points from the table to calculate the slope. For example, we can use the points (1, 5650) and (2, 5525).
$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the chosen points into the slope formula.
$m = \frac{5525 - 5650}{2 - 1}$

Calculate the slope.
$m = \frac{-125}{1} = -125$

Now that we have the slope, we need to find the y-intercept ( $b$ ). The y-intercept is the altitude when $x$ (the number of hours hiked) is 0.

To find the y-intercept, we can use the slope and one of the points from the table to plug into the point-slope form of the linear equation $y - y_1 = m(x - x_1)$ .

Choose a point from the table. For simplicity, let's use the first point (1, 5650).
$y - 5650 = -125(x - 1)$

Now, solve for $y$ to find the y-intercept.
$y = -125(x - 1) + 5650$

Distribute the slope through the parentheses.
$y = -125x + 125 + 5650$

Combine like terms to find the y-intercept.
$y = -125x + 5775$

Compare the equation we found with the students' equations.
1. Mateo's equation: $y = 125x + 5775$
2. Julie's equation: $y = -125x + 5775$
3. Oliver's equation: $y = -125x + 5650$

Identify the correct equation by matching the slope and y-intercept we calculated.
The correct equation is Julie's equation: $y = -125x + 5775$ .