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Read the description of a proportional relationship. $\newline$ Every Sunday, Jack and his friends get together to play their favorite board game, The Gems of Rawumba. Whoever collects the most gems wins! Players can earn gems by building mines. There is a proportional relationship between the number of mines Jack builds, $x$ , and the number of gems he earns, $y$ . $\newline$ If Jack builds $5$ mines, he earns $10$ gems. Write the equation for the relationship between $x$ and $y$ . $\newline$ $y = \_$

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Write and solve equations for proportional relationships

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Q. Read the description of a proportional relationship.

\newline

Every Sunday, Jack and his friends get together to play their favorite board game, The Gems of Rawumba. Whoever collects the most gems wins! Players can earn gems by building mines. There is a proportional relationship between the number of mines Jack builds,

x

, and the number of gems he earns,

y

.

\newline

If Jack builds

5

mines, he earns

10

gems. Write the equation for the relationship between

x

and

y

.

\newline

y = \_

Identify Given Values: Step $1$ : Identify the given values from the problem description. $\newline$ Jack builds $5$ mines ( $x = 5$ ) and earns $10$ gems ( $y = 10$ ). We need to find the constant of proportionality, $k$ . $\newline$ Calculation: $k = \frac{y}{x} = \frac{10}{5} = 2$ .
Calculate Constant of Proportionality: Step $2$ : Use the constant of proportionality to write the equation for the proportional relationship. $\newline$ Since $k = 2$ , the equation relating $x$ and $y$ is $y = kx$ . $\newline$ Therefore, $y = 2x$ .

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