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For each equation, determine whether \( x \) and \( y \) are directly proportional (that is, if the equation shows direct variation). If so, then find the constant of proportionality (the constant of variation). \begin{tabular}{|l|l}\hline \( \begin{array}{ll}\text { (a) } y=\frac{8}{5} x & \text { (b) } y=-x \\ \text { OProportional } \\ \text { Constant of proportionality: } k=\square & \text { OProportional } \\ \text { ONot proportional }\end{array} \)\end{tabular} \( \begin{array}{l}\text { ONot proportional of proportionality: } k=\square\end{array} \)

Ask by Kelly Murphy. in the United States
Jan 23,2025

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- **(a) \( y = \frac{8}{5} x \):** Directly proportional with \( k = \frac{8}{5} \). - **(b) \( y = -x \):** Directly proportional with \( k = -1 \).

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In equation (a) \( y = \frac{8}{5} x \), we can see that \( y \) is directly proportional to \( x \) because it can be expressed in the form \( y = kx \) where \( k = \frac{8}{5} \). Hence, the constant of proportionality is \( k = \frac{8}{5} \). In equation (b) \( y = -x \), this can also be rearranged to fit the directly proportional form \( y = kx \), where \( k = -1 \). Thus, \( y \) is directly proportional to \( x \) in a negative correlation, and the constant of proportionality is \( k = -1 \).

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