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Chapter 9: Problem 46

Use a graphing calculator to solve each system. Express solutions with approximations to the nearest thousandth. $$\begin{aligned} &0.2 x+\sqrt{2} y=1\\\ &\sqrt{5} x+0.7 y=1 \end{aligned}$$

Short Answer

Expert verified
The solution to the system is approximately (1.028, 0.336).

Step by step solution

01

Enter the equations into the graphing calculator.

Input the two equations into the graphing calculator: \(0.2 x + \sqrt{2} y = 1\) and \(\sqrt{5} x + 0.7 y = 1\).
02

Graph the equations.

Graph both equations on the calculator to find their point of intersection.
03

Find the intersection point.

Use the graphing calculator's 'intersect' feature to calculate the coordinates of the intersection point of the two lines.
04

Round the solution.

Round the intersection point coordinates to the nearest thousandth for the final solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

graphing calculator
A graphing calculator is a powerful tool designed to help plot graphs of mathematical functions. It's not just a regular calculator because it can display equations visually, which helps you understand their solutions better. For this exercise, we use a graphing calculator to input and graph the given linear equations. This visual representation makes it easier to find their intersection point, which is where the two lines cross each other. Begin by carefully entering the equations into the calculator, making sure all coefficients and constants are correct. After entering the equations, the calculator will graph the lines for you.
intersection point
Understanding the intersection point is crucial when solving systems of linear equations. The intersection point represents the coordinates \(x, y\) where the two lines meet on the graph. In simpler terms, it's the solution to the system of equations. By graphing the equations on a graphing calculator, you can visually see where the lines cross. Use the calculator's 'intersect' function, which automatically computes this point for you. This feature saves you from manual calculations, making the process quicker and less prone to error. The coordinates of this point are then used as the solution to the system.
linear equations
Linear equations are equations of the first degree that graph as straight lines. These equations can typically be written in the form \(ax + by = c\). In our exercise, we've given the system \(0.2 x + \sqrt{2} y = 1\) and \(\sqrt{5} x + 0.7 y = 1\). Each linear equation represents a line and when graphed, the intersection point of these lines represents the solution to the system. The key to solving these equations is to ensure they are entered correctly into the graphing calculator. Accurate entry of these equations helps in obtaining a precise graph, thereby a correct solution.
rounding solutions
Once you find the intersection point using the graphing calculator, the next step is to round the solution. Rounding is important to present a more manageable number. In this problem, solutions need to be rounded to the nearest thousandth, which is three decimal places. For instance, if the calculator gives you a solution like (1.2345, 2.34567), you would round each number to three decimal places, resulting in (1.235, 2.346). To round correctly, look at the fourth decimal place: if it is 5 or higher, round the third place up by one. If it's 4 or lower, keep the third place as it is. This final step ensures your answers are clean and easy to understand.

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Most popular questions from this chapter

Solve each problem. Plate-Glass Sales The amount of plate-glass sales \(S\) (in millions of dollars) can be affected by the number of new building contracts \(B\) issued (in millions) and automobiles \(A\) produced (in millions). A plate-glass company in California wants to forecast future sales by using the past three years of sales. The totals for the three years are given in the table. To describe the relationship among these variables, we can use the equation $$ S=a+b A+c B $$. where the coefficients \(a, b,\) and \(c\) are constants that must be determined before the equation can be used. (Source: Makridakis, S., and S. Wheelwright, Forecasting Methods for Management, John Wiley and Sons.) (a) Substitute the values for \(S, A,\) and \(B\) for each year from the table into the equation \(S=a+b A+c B,\) and obtain three linear equations involving \(a, b,\) and \(c\) (b) Use a graphing calculator to solve this linear system for \(a, b,\) and \(c .\) Use matrix inverse methods. (c) Write the equation for \(S\) using these values for the coefficients. $$\begin{array}{|c|c|c|} \hline S & A & B \\ \hline 602.7 & 5.543 & 37.14 \\\ \hline 656.7 & 6.933 & 41.30 \\ \hline 778.5 & 7.638 & 45.62 \\ \hline \end{array}$$ (d) For the next year it is estimated that \(A=7.752\) and \(B=47.38 .\) Predict \(S .\) (The actual value for \(S\) was \(877.6 .\) ) (e) It is predicted that in 6 yr, \(A=8.9\) and \(B=66.25 .\) Find the value of \(S\) in this situation and discuss its validity.

Supply and Demand In many applications of economics, as the price of an item goes up, demand for the item goes down and supply of the item goes up. The price where supply and demand are equal is the equilibrium price, and the resulting sup. ply or demand is the equilibrium supply or equilibrium demand. Suppose the supply of a product is related to its price by the equation $$p=\frac{2}{3} q$$ where \(p\) is in dollars and \(q\) is supply in appropriate units. (Here, \(q\) stands for quantity.) Furthermore, suppose demand and price for the same product are related by $$p=-\frac{1}{3} q+18$$ where \(p\) is price and \(q\) is demand. The system formed by these two equations has solution \((18,12),\) as seen in the graph. (GRAPH CANNOT COPY) Suppose the demand and price for a certain model of electric can opener are related by \(p=16-\frac{5}{4} q\), where \(p\) is price, in dollars, and \(q\) is demand, in appropriate units. Find the price when the demand is at each level. (a) 0 units (b) 4 units (c) 8 units

Solve each system by using the inverse of the coefficient matrix. $$\begin{array}{l} -x+y=1 \\ 2 x-y=1 \end{array}$$

Solve each problem. A sparkling-water distributor wants to make up 300 gal of sparkling water to sell for \(\$ 6.00\) per gallon. She wishes to mix three grades of water selling for \(\$ 9.00, \$ 3.00,\) and \(\$ 4.50\) per gallon, respectively. She must use twice as much of the S4.50 water as of the \(\$ 3.00\) water. How many gallons of each should she use?

Use a system of equations to solve each problem. Find the equation of the line \(y=a x+b\) that passes through the points \((-2,1)\) and \((-1,-2)\)
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