Problem 12 - Algebra Form and Function

Question

(a) Write a constraint equation. (b) Choose two solutions. (c) Graph the equation and mark your solutions. The relation between the time spent walking and the time spent canoeing on a 30 mile trip if you walk at 4 mph and canoe at 7 mph.

Step-by-Step Solution

Answer

Question: Create a constraint equation for a 30-mile trip in which a person walks 4 mph and canoes 7 mph. Find two solutions and graph the equation, marking those solutions on the graph. Answer: The constraint equation is 4x + 7y = 30, where x is the time spent walking and y is the time spent canoeing. Two solutions are (0, 4.29) and (5, 1.43). Graph the equation y = -4/7x + 30/7, and mark the solutions on the graph.

Step 1: Create the constraint equation

First, let's denote the time spent walking by x hours and the time spent canoeing by y hours. The total distance traveled must be equal to 30 miles. Since the walking speed is 4 mph and the canoeing speed is 7 mph, the equation will be: 4x + 7y = 30

Step 2: Choose two solutions

Now, we want to find two points (x, y) or two solutions for the constraint equation. We can do this by choosing a value for x (or y) and solving for the corresponding value of y (or x). For our example, we'll choose two values for x and solve for y: Solution 1: Let x = 0 (no time spent walking), 4(0) + 7y = 30 7y = 30 y = 30/7 = 4.2857 (approximately 4.29 hours spent canoeing) Solution 2: Let x = 5 (5 hours spent walking), 4(5) + 7y = 30 20 + 7y = 30 7y = 10 y = 10/7 = 1.4286 (approximately 1.43 hours spent canoeing) So, our two chosen solutions are (0, 4.29) and (5, 1.43).

Step 3: Graph the equation and mark your solutions

To graph the equation, we need to rewrite it in slope-intercept form (y = mx + b), where m is the slope, and b is the y-intercept. 4x + 7y = 30 7y = -4x + 30 y = -4/7x + 30/7 Now, we plot the line using the slope (-4/7) and y-intercept (30/7). Finally, we mark the two solutions we found in step 2 on the graph. This will give a visual representation of how the time spent walking and canoeing together completes the 30-mile trip.

Key Concepts

Distance-Speed-Time Relationship

In this exercise, we're dealing with a classic type of word problem in algebra that involves the relationship between distance, speed, and time. Understanding this concept can help us solve real-world problems easily. The distance-speed-time relationship is given by the formula:\[\text{Distance} = \text{Speed} \times \text{Time}\]This means that to find the distance traveled, you simply multiply the speed at which you're traveling by the amount of time you've been traveling. In this particular exercise, we have two scenarios: walking and canoeing.

• Walking at 4 mph
• Canoeing at 7 mph

The total distance for the trip is 30 miles. Therefore, the combination of walking and canoeing should add up to this distance. We use the distance-speed-time relationship to set up our constraint equation, which guides us in solving the problem.

Linear Equations

Linear equations are equations of the first degree, meaning they involve no exponents higher than one. They are represented in the form:\[ax + by = c\]Here, we deal with the linear equation:\[4x + 7y = 30\]This equation represents our constraint, where \(x\) is the time spent walking and \(y\) is the time spent canoeing. Linear equations are powerful tools in algebra because they can represent real-world problems like the one in this exercise. Solving these equations requires you to find values for \(x\) and \(y\) that make the equation true.In our solution:

• We found two solutions: \((0, 4.29)\) and \((5, 1.43)\).
• Each of these solutions represents a possible combination of walking and canoeing times that together add up to the 30-mile journey.

Graphing Solutions

Graphing is a visual way of solving and understanding linear equations. By plotting these equations on a graph, we can see how solutions meet the equation's conditions. To graph our equation, we need to convert it into slope-intercept form:\[y = mx + b\]The given equation is rewritten as:\[y = -\frac{4}{7}x + \frac{30}{7}\]Here, \(-\frac{4}{7}\) is the slope, and \(\frac{30}{7}\) is the y-intercept. The graph will show a straight line, and each point on this line is a solution to the equation.

• The slope tells us the rate at which one variable changes, relative to the other.
• The intercept shows where the line crosses the y-axis.

The points ((0, 4.29) and (5, 1.43)) we calculated earlier are marked on this graph, providing a visual of two plausible scenarios of walking and canoeing times.

Word Problems in Algebra

Word problems are a practical way to learn algebra because they involve using math to solve real-life situations. The key to handling word problems is translating the worded situation into mathematical expressions or equations. In this exercise, we start by identifying what is given:

• Total distance = 30 miles
• Walking speed = 4 mph
• Canoeing speed = 7 mph

Then, we decide what we are trying to find: the time spent walking and canoeing that makes up the entire trip. By creating equations based on the word problems, we can use algebra to find the solutions. It takes practice to get good at this, but each word problem solved enhances comprehension and builds the skills needed for more complex mathematics.