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Use the table to answer the question.

x
y

6 16
15 43
Write the equation of the linear function that models the relationship shown in the table. Enter your answer in y=mx+b
form.

5 answers

To find the equation of the linear function in the form \(y = mx + b\), we can use the two given points from the table: \((6, 16)\) and \((15, 43)\).

  1. Calculate the slope (m) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1) = (6, 16)\) and \((x_2, y_2) = (15, 43)\).

    Plugging in the values: \[ m = \frac{43 - 16}{15 - 6} = \frac{27}{9} = 3 \]

  2. Use the slope (m) and one of the points to find the y-intercept (b). We can use the point \((6, 16)\): \[ y = mx + b \] Substituting \(m = 3\), \(x = 6\), and \(y = 16\): \[ 16 = 3(6) + b \] \[ 16 = 18 + b \] \[ b = 16 - 18 = -2 \]

  3. Write the equation of the linear function: \[ y = 3x - 2 \]

Thus, the equation of the linear function that models the relationship shown in the table is: \[ y = 3x - 2 \]

Last year, Alayah ordered 4 cubic yards of soil and spent $220. This year, she orders 7 cubic yards of soil and spends $340. Write the equation of the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost. Enter your answer in y=mx+b
form.(1 point)

To find the equation of the linear function that models the relationship between the number of cubic yards of soil ordered (let's denote this as \( x \)) and the total cost (denote as \( y \)), we can use the two data points provided:

  1. Last year: \( (4, 220) \)
  2. This year: \( (7, 340) \)

Step 1: Calculate the Slope (m)

Using the formula for the slope \( m \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the points \( (x_1, y_1) = (4, 220) \) and \( (x_2, y_2) = (7, 340) \): \[ m = \frac{340 - 220}{7 - 4} = \frac{120}{3} = 40 \]

Step 2: Use the Slope (m) and a Point to Find the y-Intercept (b)

We can use the slope \( m = 40 \) and one of the points to find \( b \). Let's use the point \( (4, 220) \):

Substituting into the equation \( y = mx + b \): \[ 220 = 40(4) + b \] \[ 220 = 160 + b \] \[ b = 220 - 160 = 60 \]

Step 3: Write the Equation

Now we have both the slope \( m \) and the y-intercept \( b \): \[ y = 40x + 60 \]

Thus, the equation of the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost is: \[ y = 40x + 60 \]

Compare Linear Function 1 with Linear Function 2. Which one has the greater rate of change? Choose 1 for Linear Function 1; choose 2 for Linear Function 2.

Linear Function 1: y=x+8

Linear Function 2: A coordinate plane shows the x-axis ranging from negative 2 to 6 in increments of 1 and the y-axis ranging from negative 2 to 12 in increments of 1. A line with arrows at both ends joins two plotted points. The coordinates of the plotted points are as follows: left parenthesis 2 comma 6 right parenthesis and left parenthesis 4 comma 10 right parenthesis.

To compare the rates of change (slopes) of the two linear functions, we need to determine the slopes of each function.

Linear Function 1:

The equation is given as: \[ y = x + 8 \]

  • The slope \( m_1 \) here is \( 1 \) (the coefficient of \( x \)).

Linear Function 2:

We need to calculate the slope based on the two given points, \( (2, 6) \) and \( (4, 10) \).

The slope \( m_2 \) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting in the coordinates:

  • \( (x_1, y_1) = (2, 6) \)
  • \( (x_2, y_2) = (4, 10) \)

Calculating the slope: \[ m_2 = \frac{10 - 6}{4 - 2} = \frac{4}{2} = 2 \]

Comparison:

  • The slope of Linear Function 1 (from the equation \( y = x + 8 \)) is \( 1 \).
  • The slope of Linear Function 2 (calculated from the points) is \( 2 \).

Since \( 2 > 1 \), Linear Function 2 has the greater rate of change.

Conclusion:

Choose 2 for Linear Function 2.