Question

3. Which tables display linear relationships? Which display proportional relationships? Explain your reasoning,

Answer 1

A linear relationship can be modeled by the following equation:

`y=mx+b`

And if we have b = 0, we also have a proportional relationship.

So if we have a table with 4 pairs of values, we can use two of them to create our equation, then we can check if the two other points are solutions of the equation.

So we have that:

`\begin{gathered} x=1,y=30\colon \\ 30=m\cdot1+b\to b=30-m \\ \\ x=2,y=15\colon \\ 15=m\cdot2+b \\ 15=2m+(30-m) \\ 15=m+30 \\ m=15-30=-15 \\ \\ b=30-m=30-(-15)=45 \\ \\ y=-15x+45 \\ \\ x=4,y=10\colon \\ 10=-15\cdot4+45 \\ 10=-60+45 \\ 10=-15\text{ (F)} \\ \\ x=5,y=5\colon \\ 5=-15\cdot5+45 \\ 5=-75+45 \\ 5=-30\text{ (F)} \end{gathered}`

The last two points are not solution of the equation found, so the table (d) doesn't display a linear or proportional relationship.