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Problem 11

Find all solutions of the given system of equations, and check your answer graphically. $$ \begin{array}{l} 3 x-2 y=6 \\ 2 x-3 y=-6 \end{array} $$

Expert verified

The solution to the given system of equations is \(x = 6\) and \(y = 6\). This is verified graphically as the point (6,6) lies at the intersection of the two lines representing the equations.

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Chapter 3

Diet The local sushi bar serves 1 -ounce pieces of raw salmon (consisting of \(50 \%\) protein) and \(1 \frac{1}{4}\) -ounce pieces of raw tuna (40\% protein). A customer's total intake of protein amounts to \(1 \frac{1}{2}\) ounces after consuming a total of three pieces. How many of each type did the customer consume? (Fractions of pieces are permitted.)

Chapter 3

Alcohol The following table shows some data from a 2000 study on substance use among 10 th graders in the United States and Europe: $$ \begin{array}{|r|c|c|c|} \hline & \text { Used Alcohol } & \text { Alcohol-Free } & \text { Totals } \\\ \hline \text { U.S. } & x & y & 14,000 \\ \hline \text { Europe } & z & w & 95,000 \\ \hline \text { Totals } & 63,550 & 45,450 & \\ \hline \end{array} $$ a. The table leads to a linear system of four equations in four unknowns. What is the system? Does it have a unique solution? What does this indicate about the given and the missing data? b. \(T\) Given that the number of U.S. 1 Oth graders who were alcohol-free was \(50 \%\) more than the number who had used alcohol, find the missing data.

Chapter 3

\(\nabla\) Invent an interesting application that leads to a system of two equations in two unknowns with a unique solution.

Chapter 3

Purchasing (from the GMAT) Elena purchased Brand X pens for \(\$ 4.00\) apiece and Brand \(Y\) pens for \(\$ 2.80\) apiece. If Elena purchased a total of 12 of these pens for \(\$ 42.00\), how many Brand X pens did she purchase?

Chapter 3

[HINT: First eliminate all fractions and decimals; see Example \(3 .]\) $$ \begin{array}{r} -0.3 x+0.5 y=0.1 \\ 0.1 x-0.1 y=0.4 \end{array} $$

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