Question

Exercises \(57-59\), you are going on a trip to the Natural History Museum. At 9:00 A.M., you leave for the museum, which is 120 miles away. At 10: 15 A.M., you are 63 miles away from the museum. Write a linear equation that gives the distance \(d\) (in miles) from the museum in terms of the time \(t .\) Let \(t\) represent the number of minutes since 9: 00 A.M.

Short answer

The linear equation that gives the distance \(d\) (in miles) from the museum in terms of the time \(t\) (in minutes) since 9:00 A.M. is \(d = -0.76t + 120\).

Step-by-step solution

Identify the Data Points

We are given two data points. The first is at 9:00 A.M. - this is when the journey began, so \(t = 0\) and \(d = 120\) miles (the initial distance from the museum). The other point is at 10:15 A.M. - at this point, 75 minutes have passed and the distance from the museum is now 63 miles. Therefore, \(t = 75\) and \(d = 63\).

Calculate the Slope

The slope or the rate of change in a linear equation is given by the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). We can treat the times as \(x\) values and the distances as \(y\) values. So, if we let the (0, 120) be \((x_1, y_1)\) and (75, 63) be \((x_2, y_2)\), the slope becomes \(m = (63 - 120) / (75 - 0) = -57/75 = -0.76\). This slope indicates that for each minute that passes, the distance from the museum decreases by 0.76 miles.

Find the Y- Intercept

The y-intercept is the value of \(y\) when \(x\) is 0, or the initial distance to the museum. In our problem, this is 120 miles. So \(b = 120\).

Write the Linear Equation

A linear equation takes the form \(y = mx + b\). Substituting our calculated values for \(m\) and \(b\), the equation becomes \(d = -0.76t + 120\). This shows the distance \(d\) from the museum as a function of the time \(t\) in minutes.

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial when dealing with linear relationships in algebra. This form is articulated as \( y = mx + b \), where \( y \) represents the dependent variable, \( x \) is the independent variable, \( m \) denotes the slope, and \( b \) signifies the y-intercept.

The slope, \( m \), correlates to the rate at which the dependent variable changes with respect to the independent variable. In the context of a trip to the museum, if \( d \) represents the distance from the destination and \( t \) portrays the time elapsed, the slope indicates how rapidly you are approaching the museum.

The y-intercept, \( b \), is the point where the line cuts the y-axis, which translates to the initial value of the dependent variable when \( x \) is zero. If we take our museum example, it's the distance from the museum when the journey began at 9:00 A.M., hence \( b = 120 \) miles.

Rate of Change

The rate of change in a linear function characterizes how one quantity changes in relation to another. In simpler terms, it’s a measure of the 'steepness' of the line that represents the function. The calculation for rate of change is the change in the dependent variable divided by the change in the independent variable, or \( m = \frac{\Delta y}{\Delta x} \).

Applying this to the museum scenario provides us with a clear picture of your trip's progress. By calculating the negative slope, we have deduced that the distance to the museum decreases by 0.76 miles each minute. This negative rate of change means you are moving closer to your destination over time.

Real-World Application

Understanding rates of change can be quite beneficial. For example, knowing how quickly you're using fuel can help in planning pit stops on a long road trip. It could also be applied to economics, where it might represent production rates or changes in stock prices over time.

Linear Equation Word Problems

Linear equation word problems involve finding unknown values using linear relationships in real-world scenarios. They typically require translating the given information into a mathematical model.

In our example exercise, the word problem provided two data points that helped us compose the linear equation \( d = -0.76t + 120 \). This equation models the distance remaining to the museum as time progresses. Solving word problems usually involves identifying key information, determining variables, and making use of linear equations to find solutions.

Steps for Success

• Read the problem thoroughly.
• Identify and define the variables.
• Determine what the problem is asking you to find.
• Translate the words into an equation using the slope-intercept form.
• Solve for the unknown variable.
• Verify your solution within the context of the problem.

By mastering these steps, students can approach linear equation word problems with greater confidence and precision.