Lisa collect rain water in a large barrel that weighs 25 pounds when there are 10 gallons of water in the barrel the total weight of the barrel and the water is 108.4 pounds when there are 20 gallons of water the total weight is 191.8 pounds which equation match this situation

We can model this situation with a linear equation. We know for our problem that the initial point is (0, 25), and the next point is (10, 108.4). To relate those points and find the slope of our linear equation, we are going to use the formula: [tex]m= \frac{y_{2}-y_{1}}{x_{2}-x_{1} } [/tex]. Notice that we know from our points that: [tex]x_{1}=0[/tex], [tex]y_{1}=25[/tex], [tex]x_{2}=10[/tex], and [tex]y_{2}=108.4[/tex]. So, lets replace those values in our formula to find [tex]m[/tex]:
[tex]m= \frac{108.4-25}{10-0} [/tex]
[tex]m= \frac{83.4}{10} [/tex]
[tex]m=8.34[/tex]

Now that we have the slope of our linear equation, we can use the point slope formula: [tex]y-y_{1}=m(x-x_{1})[/tex] to complete our linear equation:
[tex]y-25=8.34(x-0)[/tex]
[tex]y-25=8.34x[/tex]
[tex]y=8.34x+25[/tex]
Or in function notation: [tex]f(x)=8.34x+25[/tex]

We can conclude that the equation that matches this situation is [tex]y=8.34x+25[/tex].