Question

the larsons are filling their swimming pool. this table shows the amount of water in gallons available in the swimming pool over time.

time (hours)	water (gallons)
5	420
8	642
12	938

which equation models the amount of water in the swimming pool, y, after x hours?

• $y = 2x + 198$
• $y = 222x - 246$
• $y = 74x$
• $y = 74x + 50$

Explanation:

Step1: Recall linear equation form

The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. To find the correct equation, we can calculate the slope between two points and then check the equations.

Let's take two points from the table, say \((x_1,y_1)=(2,198)\) and \((x_2,y_2)=(5,420)\). The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{420 - 198}{5 - 2}=\frac{222}{3} = 74\).

Step2: Check the equations with slope \( m = 74\)

Now we check the equations with \( m = 74\). The equations are:

• \( y=2x + 198\) (slope \( m = 2\), not 74, so eliminate)
• \( y = 222x-246\) (slope \( m=222\), not 74, eliminate)
• \( y = 74x\) (let's check with \( x = 2\): \( y=74\times2=148

eq198\))

• \( y=74x + 50\) (check with \( x = 2\): \( y=74\times2+50=148 + 50=198\) (matches). Check with \( x = 5\): \( y=74\times5+50=370+50 = 420\) (matches). Check with \( x = 8\): \( y=74\times8+50=592+50=642\) (matches). Check with \( x = 12\): \( y=74\times12+50=888+50 = 938\) (matches))

Answer:

\( y = 74x + 50\) (the option with this equation)