Every Sunday, Barbara and her 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 Barbara builds, x , an the number of gems she earns, y . D) x (mines) & D) y (gems) 2 & 6 3 & 9 5 & 15 6 & 18 Write an equation for the relationship between x and y . Simplify any fractions. [ y= ]

## Step 1: Understand the relationship between \( x \) and \( y \)<br />### The problem states that there is a proportional relationship between the number of mines (\( x \)) and the number of gems (\( y \)). This means that \( y \) is directly proportional to \( x \), which can be expressed as:<br />\[<br />y = kx<br />\]<br />where \( k \) is the constant of proportionality.<br /><br />## Step 2: Determine the constant of proportionality (\( k \))<br />### To find \( k \), we use one pair of values from the table. For example, when \( x = 2 \) and \( y = 6 \):<br />\[<br />k = \frac{y}{x} = \frac{6}{2} = 3<br />\]<br /><br />## Step 3: Write the equation<br />### Substituting \( k = 3 \) into the proportional relationship \( y = kx \), we get:<br />\[<br />y = 3x<br />\]<br /><br />## Step 4: Verify the equation<br />### To ensure the equation \( y = 3x \) works for all data points in the table, substitute each \( x \)-value and check if it produces the corresponding \( y \)-value:<br />- For \( x = 2 \): \( y = 3(2) = 6 \) (correct)<br />- For \( x = 3 \): \( y = 3(3) = 9 \) (correct)<br />- For \( x = 5 \): \( y = 3(5) = 15 \) (correct)<br />- For \( x = 6 \): \( y = 3(6) = 18 \) (correct)<br /><br />### Since the equation holds true for all data points, it is correct.