2.3.1: Systems of Linear Equations – Special Cases (Exercises)

1. \begin{aligned}
   2 x+6 y&=8 \\
   x+3 y&=4
   \end{aligned}

2. The sum of digits of a two digit number is 9. The sum of the number and the number obtained by interchanging the digits is 99. Find the number.

3. \begin{aligned}
   2 x-y &=10 \\
   -4 x+2 y &=15
   \end{aligned}

4. \begin{aligned}
   x+y+z&=6 \\
   3 x+2 y+z&=14 \\
   4 x+3 y+2 z&=20
   \end{aligned}

5. \begin{aligned}
   x+2 y-4 z&=1 \\
   2 x-3 y+8 z&=9
   \end{aligned}

6. Jessica has a collection of 15 coins consisting of nickels, dimes and quarters. If the total worth of the coins is $1.80, how many are there of each? Find all three solutions.

7. A company is analyzing sales reports for three products: products X, Y, Z. One report shows that a combined total of 20,000 of items X, Y, and Z were sold. Another report shows that the sum of the number of item Z sold and twice the number of item X sold equals 10,000. Also item X has 5,000 more items sold than item Y. Are these reports consistent?

8. \begin{aligned}
   x+y+2 z=0 \\
   x+2 y+z=0 \\
   2 x+3 y+3 z=0
   \end{aligned}

9. Find three solutions to the following system of equations.

\begin{aligned}
x+2 y+z &=12 \\
y &=3
\end{aligned}

10. \begin{aligned}
    x+2 y&=5 \\
    2 x+4 y&=k
    \end{aligned}

For what values of k does this system of equations have

1. No solution?
2. Infinitely many solutions?

11. \(x + 3y - z = 5\)

12. Why is it not possible for a linear system to have exactly two solutions? Explain geometrically.