The difference of two complementary angles is 26 degrees. Identify what we are looking for: We are looking for the speed of the ship in still water and the speed of the current. As pointless and repetitive as the exercises are, the feeble attempts by the textbook authors to make the problems relevant are worse. The wife earns $16,000 less than twice what her husband earns. We also know the final volume is 42 gallons. A Real World Dilemma! The point at which the two lines intersect is called the break-even point. A wind current blowing against the direction of the plane is called a headwind. We’ll do another example where we stop after we write the system of equations. Mark burned 11 calories for each minute of yoga and 7 calories for each minute of jumping jacks. Define and Translate: We will call the unknown volume of the 24% solution x, and the unknown volume of the 18% solution y. The concentration for this amount is 0.6 because we want the final solution to be 60% methane. The total mass of the final solution comes from, When you sum the amount column you get one equation: [latex]x+ y = 42[/latex] \(\left\{\begin{array}{l}{w+h=84,000} \\ {h=2 w-18,000}\end{array}\right.\). How many skateboards must be produced and sold before a profit is possible? The sum of their ages is 50. Find the measures of the angles. Solutions used for most purposes typically come in pre-made concentrations from manufacturers, so if you need a custom concentration, you would need to mix two different strengths. [latex]\begin{array}{c}c+a=2,000\\ 25c+50a=70,000\end{array}[/latex]. It will take Sally \(1\frac{1}{2}\) hours to catch up to Charlie. Find the speed of the ship in still water and the speed of the river current. There are many such applications for linear equations. \\ \\ {\textbf{Step 3. The angle measures are 55 degrees and 35 degrees. Now, the boat is going upstream, opposite to the river current. Below, we summarize three key factors that will help guide you in translating a situation into a system. Cross-country airplane flights in the United States generally take longer going west than going east because of the prevailing wind currents. Either by looking at the graph, or noting that [latex]K\left(d\right)[/latex] is growing at a slower rate, we can conclude that Keep on Trucking, Inc. will be the cheaper price when more than 100 miles are driven, that is [latex]d>100[/latex]. This is our equation for finding the unknown volume. The current helps push the boat, so the boat’s actual speed is faster than its speed in still water. The break-even point is [latex]\left(50,000,77,500\right)[/latex]. The concentration or strength of a liquid solution is often described as a percentage. Find the speed of the jet in still air and the speed of the wind. How many calories were burned for each minute of yoga? Let c = the number of children and a = the number of adults in attendance. One application of systems of equations are mixture problems. The speed of the canoe is 7 mph and the speed of the current is 1 mph. Substitute the expression [latex]2,000-c[/latex] in the second equation for [latex]a[/latex] and solve for [latex]c[/latex]. Depending on which way the boat is going, the current of the water is either slowing it down or speeding it up. Find the speed of the jet in still air and the speed of the wind. The cost function is the function used to calculate the costs of doing business. [latex]\begin{array}{l}P\left(x\right)=1.55x-\left(0.85x+35,000\right)\hfill \\ \text{ }=0.7x - 35,000\hfill \end{array}[/latex]. How many calories does she burn for each minute of circuit training? [latex]\begin{array}{cc} (14) + y &= 42\\ y &= 28 \end{array}[/latex]. Let s=s= the rate of the ship in still water. Since rate times time is distance, we can write the system of equations. The ship goes downstream and then upstream. The [latex]x[/latex] -axis represents quantity in hundreds of units. In this section, we will practice writing equations that represent the outcome from mixing two different concentrations of solutions. There are many other disciplines that use solutions as well. The actual speed of the boat is [latex]b-c[/latex]. The husband earns $18,000 less than twice what the wife earns. The shaded region to the left represents quantities for which the company suffers a loss. Exercise \(\PageIndex{1}\): How to Translate to a System of Equations. The basic equation was D = rt where D is the distance traveled, r is the rate, and t is the time. Two angles are complementary if the sum of the measures of their angles is 90 degrees. \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), 5.4: Solve Applications with Systems of Equations, [ "article:topic", "license:ccby", "showtoc:no", "authorname:openstaxmarecek" ], \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\). Find the length and width of the dog run. [latex]\begin{array}{c}0.85x+35,000=1.55x\\ 35,000=0.7x\\ 50,000=x\end{array}[/latex], Then, we substitute [latex]x=50,000[/latex] into either the cost function or the revenue function. Here’s a “real world” example of linear equations: You and your friend together sell 58 tickets to a raffle. Using the rates of change and initial charges, we can write the equations, [latex]\begin{array}{l}K\left(d\right)=0.59d+20\\ M\left(d\right)=0.63d+16\end{array}[/latex]. There are two long sides and the one shorter side is parallel to the house. Define your variables. Identify}\text{ what we are looking for.}} The length is 85 feet and the width is 20 feet. The angle measures are 5 degrees and 85 degrees. The profit function is found using the formula [latex]P\left(x\right)=R\left(x\right)-C\left(x\right)[/latex].