Modeling with Linear Equations

A pool contains 78 gallons of water and needs more water added. Water is added at a rate of 3 gallons per minute.  Which equation represents the total gallons T of the water in the pool after m minutes?

Answer explanation

final amount = beginning amount + rate of change multiplied by time

• When the equation is solved for y, it is easy to determine how many pairs of mittens can be bought if we know how many hats are bought.

• When the equation is solved for y, it is easy to determine how many hats can be bought if we know how many pairs of mittens are bought.

Answer explanation

It is helpful to write the order of the items in these types of problems where x and y are used as the variables.

In this problem, the order is (mittens, hats).

So when the equation is written y=____, it's the same as mittens=____.

• $100 spent on hats and $125 spent on mittens

• 100 hats purchased and 125 pairs of mittens purchased

• 125 hats purchased and 100 pairs of mittens purchased

• $100 spent on mittens and $125 spent on hats

Answer explanation

It is helpful to write the order of the items when the variables are x and y.

In this problem, the order is (mittens, hats).

Answer explanation

You can always find the y-intercept by setting x=0.

In this case,

y=17/2 + 3/2 times 0

y=17/2 + 0

y=17/2

Got it! (0,17/2)

Answer explanation

Remember that when the equation is solved for y, then the slope will be the coefficient on x.

The negative sign on the 7/5 is showing us the coefficient on x is -7/5, so the slope is NEGATIVE 7/5.

slope = -7/5

Answer explanation

Remember that when the equation is solved for y, then the slope will be the coefficient on x.

The coefficient is the number multiplied with the x.

• The coefficient of the x is the slope of the line.

• This equation doesn't have a slope, since there is no fraction.

Answer explanation

The slope is sometimes a fraction -- but not always.

When the equation is solved for y, the slope is always the coefficient of the x.

So this equation will have slope of 4.

Select all true statements.

• The slope of a line is the change in y divided by the change in x.

• The slope of a line is the rate of change.

• The slope of a line is the vertical change divided by the horizontal change.

• The slope of a line is the run divided by the rise.

Answer explanation

The incorrect choice says the slope is the run/rise,

but the slope is the rise divided by the run.

A jet ski rental charges a fee of $50 for rental insurance and $25 per hour to rent a jet ski. Which equation represents this?

Answer explanation

final amount = beginning amount + rate of change multiplied by time

This graph represents the number of brocolli and tomato plants that will use all available space in a garden.

Select all true statements.

• Three brocolli plants could be replaced with 2 tomato plants and all available space would still be used.

• Four brocolli plants could be replaced with 6 tomato plants and all available space would still be used.

Answer explanation

The slope of -8/12 brocolli plants/tomato plants shows us that you can change out

8 brocolli plants for 12 tomato plants or

4 brocolli plants for 6 tomato plants or

2 brocolli plants for 3 tomato plants or

3 tomato plants for 2 brocolli plants etc.