- Find the point which lies on both lines, x + 3y = 1 and 4x − y = 3.
- Solve Problem 1 graphically. That is, graph each line and see where they intersect.
- Find the point of intersection of the two lines 3x + y = 3 and x + 2y = 1.
- Solve Problem 3 graphically. That is, graph each line and see where they intersect.
- Do the three lines, x+2y = 1,2x−y = 1, and 4x+3y = 3 have a common point of intersection? If so, find the point and if not, tell why they don’t have such a common point of intersection.
- Do the three planes, x + y − 3z = 2, 2x + y + z = 1, and 3x + 2y − 2z = 0 have a common point of intersection? If so, find one and if not, tell why there is no such point.
- You have a system of k equations in two variables, k ≥ 2. Explain the geometric significance
of
- No solution.
- A unique solution.
- An infinite number of solutions.

- Here is an augmented matrix in which ∗ denotes an arbitrary number and ■ denotes a nonzero
number. Determine whether the given augmented matrix is consistent. If consistent, is the solution
unique?
- Here is an augmented matrix in which ∗ denotes an arbitrary number and ■ denotes a nonzero
number. Determine whether the given augmented matrix is consistent. If consistent, is the solution
unique?
- Here is an augmented matrix in which ∗ denotes an arbitrary number and ■ denotes a nonzero
number. Determine whether the given augmented matrix is consistent. If consistent, is the solution
unique?
- Here is an augmented matrix in which ∗ denotes an arbitrary number and ■ denotes a nonzero
number. Determine whether the given augmented matrix is consistent. If consistent, is the solution
unique?
- Suppose a system of equations has fewer equations than variables. Must such a system be consistent? If so, explain why and if not, give an example which is not consistent.
- If a system of equations has more equations than variables, can it have a solution? If so, give an example and if not, tell why not.
- Find h such that
is the augmented matrix of an inconsistent matrix.

- Find h such that
is the augmented matrix of a consistent matrix.

- Find h such that
is the augmented matrix of a consistent matrix.

- Choose h and k such that the augmented matrix shown has one solution. Then choose h and k such
that the system has no solutions. Finally, choose h and k such that the system has infinitely many
solutions.
- Choose h and k such that the augmented matrix shown has one solution. Then choose h and k such
that the system has no solutions. Finally, choose h and k such that the system has infinitely many
solutions.
- Determine if the system is consistent. If so, is the solution unique?
- Determine if the system is consistent. If so, is the solution unique?
- Find the general solution of the system whose augmented matrix is
- Find the general solution of the system whose augmented matrix is
- Find the general solution of the system whose augmented matrix is
- Find the general solution of the system whose augmented matrix is
- Find the general solution of the system whose augmented matrix is
- Give the complete solution to the system of equations, 7x + 14y + 15z = 22, 2x + 4y + 3z = 5, and 3x + 6y + 10z = 13.
- Give the complete solution to the system of equations, 3x − y + 4z = 6, y + 8z = 0, and −2x + y = −4.
- Give the complete solution to the system of equations,9x− 2y + 4z = −17, 13x− 3y + 6z = −25, and −2x − z = 3.
- Give the complete solution to the system of equations,65x + 84y + 16z = 546, 81x + 105y + 20z = 682, and 84x + 110y + 21z = 713.
- Give the complete solution to the system of equations, 8x + 2y + 3z = −3,8x + 3y + 3z = −1, and 4x + y + 3z = −9.
- Give the complete solution to the system of equations, −8x + 2y + 5z = 18,−8x + 3y + 5z = 13, and −4x + y + 5z = 19.
- Give the complete solution to the system of equations, 3x − y − 2z = 3, y − 4z = 0, and −2x + y = −2.
- Give the complete solution to the system of equations, −9x + 15y = 66,−11x + 18y = 79 ,−x + y = 4, and z = 3.
- Give the complete solution to the system of equations, −19x + 8y = −108, −71x + 30y = −404, −2x + y = −12, 4x + z = 14.
- Consider the system −5x + 2y − z = 0 and −5x − 2y − z = 0. Both equations equal zero and so −5x + 2y − z = −5x − 2y − z which is equivalent to y = 0. Thus x and z can equal anything. But when x = 1, z = −4, and y = 0 are plugged in to the equations, it doesn’t work. Why?
- Four times the weight of Gaston is 150 pounds more than the weight of Ichabod. Four times the weight of Ichabod is 660 pounds less than seventeen times the weight of Gaston. Four times the weight of Gaston plus the weight of Siegfried equals 290 pounds. Brunhilde would balance all three of the others. Find the weights of the four sisters.
- The steady state temperature, u in a plate solves Laplace’s equation, Δu = 0. One way to
approximate the solution which is often used is to divide the plate into a square mesh and require the
temperature at each node to equal the average of the temperature at the four adjacent nodes. This
procedure is justified by the mean value property of harmonic functions. In the following picture, the
numbers represent the observed temperature at the indicated nodes. Your task is to find the
temperature at the interior nodes, indicated by x,y,z, and w. One of the equations is
z = .
- Consider the following diagram of four circuits.
Those jagged places denote resistors and the numbers next to them give their resistance in ohms, written as Ω. The breaks in the lines having one short line and one long line denote a voltage source which causes the current to flow in the direction which goes from the longer of the two lines toward the shorter along the unbroken part of the circuit. The current in amps in the four circuits is denoted by I

_{1},I_{2},I_{3},I_{4}and it is understood that the motion is in the counter clockwise direction. If I_{k}ends up being negative, then it just means the current flows in the clockwise direction. Then Kirchhoff’s law states thatThe sum of the resistance times the amps in the counter clockwise direction around a loop equals the sum of the voltage sources in the same direction around the loop.

In the above diagram, the top left circuit should give the equation

For the circuit on the lower left, you should have

Write equations for each of the other two circuits and then give a solution to the resulting system of equations. You might use a computer algebra system to find the solution. It might be more convenient than doing it by hand.

- Consider the following diagram of three circuits.
Those jagged places denote resistors and the numbers next to them give their resistance in ohms, written as Ω. The breaks in the lines having one short line and one long line denote a voltage source which causes the current to flow in the direction which goes from the longer of the two lines toward the shorter along the unbroken part of the circuit. The current in amps in the four circuits is denoted by I

_{1},I_{2},I_{3}and it is understood that the motion is in the counter clockwise direction. If I_{k}ends up being negative, then it just means the current flows in the clockwise direction. Then Kirchhoff’s law states thatThe sum of the resistance times the amps in the counter clockwise direction around a loop equals the sum of the voltage sources in the same direction around the loop. Find I

_{1},I_{2},I_{3}. - Here are some chemical reactions. Balance them.
- KNO
_{3}+ H_{2}CO_{3}→ K_{2}CO_{3}+ HNO_{3} - Ba
_{3}N_{2}+ H_{2}O → Ba_{2}+ NH_{3} - CaCl
_{2}+ Na_{3}PO_{4}→ Ca_{3}_{2}+ NaCl

- KNO
- In the section on dimensionless variables 200 it was observed that ρV
^{2}AB has the units of force. Describe a systematic way to obtain such combinations of the variables which will yield something which has the units of force.

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