MAT137 Week 3: Solving Systems of Equations

Welcome to Week 3 of MAT137! This week, we're tackling systems of equations. These are sets of two or more equations containing two or more variables. Our goal is to find the values of these variables that satisfy all equations in the system simultaneously. We will explore systems involving two and three variables.

Key Concepts and Methods

We'll be focusing on three primary methods for solving these systems:

  • Graphing: This method involves plotting each equation on a coordinate plane. The solution to the system is the point (or points) where the graphs intersect. This is particularly useful for visualizing the solutions of systems of two equations with two variables. When using graphing, pay close attention to the scales and intercepts for accuracy. Also, consider the slopes of the lines. Parallel lines indicate no solution. Identical lines indicate infinitely many solutions.
  • Substitution: In this method, we solve one equation for one variable in terms of the other(s). Then, we substitute this expression into the other equation(s). This eliminates one variable, allowing us to solve for the remaining variable(s). For example, in the system $x + y = 5$ and $2x - y = 1$, we could solve the first equation for $y$ to get $y = 5 - x$. Then substitute that into the second equation: $2x - (5 - x) = 1$. This simplifies to $3x - 5 = 1$, and then $3x = 6$, so $x = 2$. Finally, substitute $x = 2$ back into $y = 5 - x$ to get $y = 3$. The solution is $(2, 3)$.
  • Elimination: This method involves manipulating the equations (multiplying by constants) so that when we add or subtract the equations, one of the variables is eliminated. This again leaves us with an equation in one variable, which we can solve. Consider the system $3x + 2y = 7$ and $4x - 2y = 0$. Notice the $y$ terms have opposite signs and the same coefficient. Add the equations together: $(3x + 2y) + (4x - 2y) = 7 + 0$, which simplifies to $7x = 7$, so $x = 1$. Substitute $x = 1$ into either equation to solve for $y$. Using the first equation: $3(1) + 2y = 7$, so $2y = 4$, and $y = 2$. The solution is $(1, 2)$.

When working with three variables, the concepts remain the same, but the process can become more involved. For instance, with elimination, you might need to eliminate one variable from two pairs of equations to end up with two equations in two variables, which you can then solve as above.

Example Problems

Let's consider the following system:

$$ \begin{cases} x + y + z = 6 \\ 2x - y + z = 3 \\ x + 2y - z = 2 \end{cases} $$

You can use elimination or substitution (or a combination!) to solve for x, y, and z. Practice is key! Try eliminating $z$ first!

Important Notes

  • Not all systems have a unique solution. Some systems may have no solutions (inconsistent systems), while others may have infinitely many solutions (dependent systems).
  • Always check your solutions by substituting them back into the original equations to ensure they satisfy all equations.

Resources

Remember to review the class notes from 11-02-04, Sections 3-2, 3-3, and 7-6 for more detailed explanations and examples.

Quiz

Don't forget to complete this week's quiz! Week 3 Quiz. Good luck, and keep practicing!