Question

writing linear equations word problems
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a lake near the arctic circle is covered by a 2 - meter - thick sheet of ice during the cold winter months. when spring arrives, the warm air gradually melts the ice, causing its thickness to decrease at a constant rate. after 3 weeks, the sheet is only 1.25 meters thick.
let y represent the ice sheets thickness (in meters) after x weeks.
complete the equation for the relationship between the thickness and number of weeks.

Explanation:

Step1: Find the rate of change (slope)

The initial thickness is $y_1 = 2$ meters at $x_1=0$ weeks, and after $x_2 = 3$ weeks, the thickness is $y_2=1.25$ meters. The slope $m$ of the linear - equation $y=mx + b$ is given by the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. So, $m=\frac{1.25 - 2}{3-0}=\frac{- 0.75}{3}=-0.25$.

Step2: Find the y - intercept

The y - intercept $b$ is the value of $y$ when $x = 0$. Here, when $x = 0$, the ice - sheet thickness $y=2$ meters, so $b = 2$.

Step3: Write the linear equation

The linear equation in slope - intercept form is $y=mx + b$. Substituting $m=-0.25$ and $b = 2$ into the equation, we get $y=-0.25x + 2$.

Answer:

$y=-0.25x + 2$