Math Problem Solver

1. Write down a system of two linear equations in two unknowns which has no solution.

Draw a picture of the situation.

2. Suppose you have a system of two linear equations in three unknowns. If a solution

exists, how many are there? What might the set of solutions look like geometrically?

3. Solve the following system of equations and describe in words each step you use:

x + 3y − z = 1,

3x + 4y − 4z = 7,

3x + 6y + 2z = −3.

How many solutions are there, and what does the solution set look like geometrically?

4. Find all solutions of the system

x + y − 3z = −5,

−5x − 2y + 3z = 7,

3x + y − z = −3.

Describe (but don’t draw) the graphs of each of the three above equations and their

intersection.

5. What condition on a, b, c, and d will guarantee that there will be exactly one solution

to the following system?

ax + by = 1,

cx + dy = 0.

6. Set up a system of linear equations for the following problem and then solve it:

The three-digit number N is equal to 15 times the sum of its digits. If you reverse the

digits of N, the resulting number is larger by 396. Also, the units (ones) digit of N is

one more than the sum of the other two digits. Find N.

7. Consider the system of equations

ax + by = k,

cx + dy = l,

ex + fy = m.

Show that if this system has a solution, then at least one equation can be thrown out

without altering the solution set slobe them