The boat is going downstream, in the same direction as the river current. A Real World Dilemma! Now let’s practice putting these key factors to work. When she spent 20 minutes on the elliptical trainer and 30 minutes circuit training she burned 473 calories. How many skateboards must be produced and sold before a profit is possible? This is a uniform motion problem and a picture will help us visualize. We’ll call the speed of the boat in still water b and the speed of the river current c. In Figure \(\PageIndex{1}\) the boat is going downstream, in the same direction as the river current. Answer}\text{ the question.} Use your knowledge of solutions of systems of linear equations to solve a real world problem you might have already been … The measures of two supplementary angles add to 180 degrees. The current helps push the boat, so the boat’s actual speed is faster than its speed in still water. A couple has a total household income of $84,000. Let c = the number of children and a = the number of adults in attendance. \\& \left\{\begin{array}{l}{x+y=90} \\ \underline{x-y=26}\end{array}\right. The angle measures are 55 degrees and 35 degrees. Let [latex]c=[/latex] the rate of the current. The downstream rate would be: Answer the question. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. The first, Keep on Trucking, Inc., charges an up-front fee of $20, then 59 cents a mile. Together they make $43 per hour. USE A PROBLEM SOLVING STRATEGY FOR SYSTEMS OF LINEAR EQUATIONS. \\ \\ {\textbf{Step 3. It is rare to be given equations that neatly model behaviors that you encounter in business, rather, you will probably be faced with a situation for which you know key information as in the example above. Alessio rides his bike \(3\frac{1}{2}\) hours at a rate of 10 miles per hour. One application of systems of equations are mixture problems. Half an hour later, Kelly left St. Louis on the same route as Joni, driving 78 miles per hour. Let s=s= the rate of the ship in still water. The length is 60 feet and the width is 35 feet. Substitute [latex]c=1,200[/latex] into the first equation to solve for [latex]a[/latex]. Have questions or comments? Find the numbers. We now have a system of linear equations in two variables. These graphs are sketched above, with K(d) in blue. Figure \(\PageIndex{1}\) and Figure \(\PageIndex{2}\) show how a river current affects the speed at which a boat is actually traveling. If two angles are supplementary, we say that one angle is the supplement of the other. A solution is a mixture of two or more different substances like water and salt or vinegar and oil. One application of systems of equations are mixture problems. Find the numbers. Sometimes it was a bit of a challenge figuring out how to name the two quantities, wasn’t it? The second, Move It Your Way, charges an up-front fee of $16, then 63 cents a mile. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for 20 minutes, her fitness app says she burned 278 calories. Multiply amount by concentration to get total, be sure to distribute on the last row: [latex]\left(70 + x\right)0.6[/latex]Add the entries in the amount column to get final amount. We can call our unknown amount x. Legal. The revenue from all children can be found by multiplying $25.00 by the number of children, [latex]25c[/latex]. A farmer has two types of milk, one that is 24% butterfat and another which is 18% butterfat. How many children and how many adults bought tickets? Define and Translate: Solution 1 is the 70 mL of 50% methane and solution 2 is the unknown amount with 80% methane. How long will it take Kelly to catch up to Joni? How many calories were burned for each minute of yoga? Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Find the measures of both angles. The sum of twice a number and nine is 31. Read the problem: This is a uniform motion problem and a picture will help us visualize the situation. Next we add the new volumes and new masses. Using what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. Find the speed of the jet in still air and the speed of the wind. This is our equation for finding the unknown volume. A river cruise ship sailed 60 miles downstream for 4 hours and then took 5 hours sailing upstream to return to the dock. In other words, the company breaks even if they produce and sell 700 units. The actual speed at which the boat is moving is [latex]b+c[/latex]. Depending on which way the boat is going, the current of the water is either slowing it down or speeding it up. The break-even point is [latex]\left(50,000,77,500\right)[/latex]. We are looking for the length of time Kelly, Substitute k=j−12 into the second equation, then solve for. This tells us that the cost from the two companies will be the same if 100 miles are driven. In this activity, students answer this question as they experience a real-world application of systems of equations. Solving equations with two variables Find the speed of the ship in still water and the speed of the river current. In our next example, we help answer the question, “Which truck rental company will give the best value?”. A chart will help us organize the information. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. & {\text{Let x = the measure of the first angle.} A diagram is useful in helping us visualize the situation. In this section, we will practice writing equations that represent the outcome from mixing two different concentrations of solutions. [latex]\begin{array}{c}4s+4c=60 \\ 5s-5c = 60\end{array}[/latex]. The images below show how a river current affects the speed at which a boat is actually travelling. If we use the skateboard example as a model, x would represent the number of frames produced (instead of skateboards) and y would represent the amount of money it would cost to produce them (the same as the skateboard problem).

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