The depth of a pond is 180 \(\mathrm{cm}\) and is being reduced by 1 \(\mathrm{cm}\) per week. The depth of a second pond is 160 \(\mathrm{cm}\) and is being reduced by \(\frac{1}{2} \mathrm{cm}\) per week. If the depths of both ponds continue to be reduced at these constant rates, in about how many weeks will the ponds have the same depth? F. 10 G. 20 H. 40 J. 80 K. 140

Understanding linear equations is crucial when studying how variables interact in relationships that are directly proportional. A linear equation can usually be written in the form of y = mx + b, where m represents the slope or rate of change, and b represents the y-intercept, the point where the line crosses the y-axis.

In the context of the ACT Math problem, we had two linear equations representing the depths of the ponds as they decrease over time. The first pond's depth equation is y = 180 - x, and the second pond's depth equation is y = 160 - 0.5x.

To visualize it, imagine plotting each pond's depth on a graph over time. The point where the two lines intersect indicates the point in time when both ponds will have the same depth. In simpler terms, we're looking for the point where both situations are equal, hence the setting of one equation equal to the other to find the x-value, which stands for weeks in this case.

Interpreting Rate of Change

The rate of change is a critical concept that represents the velocity at which one quantity changes in relation to another. In algebra, the rate of change is synonymous with the slope of a line and can tell us how steep a line is on a graph. The higher the rate, the steeper the line.

In our problem, the first pond's rate of change is 1 cm/week, represented by the slope of the line. For the second pond, the rate is 0.5 cm/week. These rates help us understand how quickly or slowly each pond's depth is decreasing, and by setting the depths equal to each other, we are actually comparing the rates to determine when they produce the same outcome.

Steps to Solve Equations

Equation solving is a foundational skill in algebra and involves finding the value(s) of the variable(s) that satisfy the equation. The goal is to isolate the variable on one side of the equation to discover its value.

For the ACT Math exercise, the equation 180 - x = 160 - 0.5x needed to be solved for the variable x, representing weeks. The solution involves combining like terms and isolating the variable. It's essential to perform the same operation on both sides of the equation to maintain equality, which ultimately leads to the answer.

By mastering equation solving, students can tackle a vast array of problems in algebra and beyond, leading to a better understanding of how variables relate to one another in mathematical models.