Questions: Solve for y - STAAR Problems Google Form Due: Fri Sep 13, 2024 11:59pm 50 The table represents some points on the graph of a linear function. x y -20 -268 -14 -196 -8 -124 -1 -40 Which equation represents the same relationship? F y+268=(1/12)(x+20) G y+20=(1/12)(x+268) H y+268=12(x+20) ] y+20=12(x+268)

Solution Steps

To find the equation that represents the relationship given by the table, we need to determine the slope and y-intercept of the linear function. We can use the formula for the slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \) using any two points from the table. Once we have the slope, we can use the point-slope form of a linear equation \( y - y_1 = m(x - x_1) \) to find the equation. Finally, we compare the derived equation with the given options to find the correct one.

To find the slope \( m \) of the linear function, we use the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points \((-20, -268)\) and \((-14, -196)\):

\[ m = \frac{-196 - (-268)}{-14 - (-20)} = \frac{72}{6} = 12.0 \]

Using the slope \( m = 12.0 \) and the point \((-20, -268)\), we can find the y-intercept \( b \) using the point-slope form of the equation:

\[ y - y_1 = m(x - x_1) \]

Substituting the values:

\[ -268 = 12.0(-20) + b \]

Solving for \( b \):

\[ -268 = -240 + b \implies b = -28.0 \]

The equation of the line in slope-intercept form is:

\[ y = mx + b \]

Substituting the values of \( m \) and \( b \):

\[ y = 12.0x - 28.0 \]

We compare the derived equation \( y = 12.0x - 28.0 \) with the given options:

• Option F: \( y + 268 = \frac{1}{12}(x + 20) \)
• Option G: \( y + 20 = \frac{1}{12}(x + 268) \)
• Option H: \( y + 268 = 12(x + 20) \)
• Option I: \( y + 20 = 12(x + 268) \)

Rewriting Option H:

\[ y + 268 = 12(x + 20) \implies y = 12x + 240 - 268 \implies y = 12x - 28 \]

This matches the derived equation.

Final Answer