2.1 Linear Systems of Equations - Notes PDF

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2.1 LINEAR SYSTEMS OF EQUATIONS A linear system of equations is a set of two more linear equations that have the same variables. A linear equation in two variables has the form ax + by = c, where a, b, and c are real constants. A linear equation in n variables has the same form, except that there are n variables with n constants on the left side of the equation. For example,

A solution to a linear system in two variables is a tuple (a pair in the case of two variables) that satisfies each equation in the system. If the system is in two variables, its solution is a pair (x, y) that satisfies every equation in the system. For example, the solution to the system shown before is (1, −2) since 1. (plug into first equation)

2. (plug into second equation)

1

We can also visualize a system of linear equations as a set of (possibly) intersecting lines on a coordinate plane. Let’s consider a linear system of two equations in two variables. There are three possible pictures we can see if we graph two linear equations in two variables: 1. independent system

2. inconsistent system

3. dependent system

The picture below shows these three possibilities:

Methods for solving a linear system of equations There are two algebraic methods for solving a system of linear equations: 1. The Substitution Method 2. The Elimination Method

The Substitution Method 1. Solve one equation for one variable. 2. Plug the resulting expression into the unused equation. 3. Solve for the variable remaining. 4. Plug the solution of one variable into any equation to solve for the other. 5. Write solution as a tuple (x1 , x2 , ..., xn ).

Example. Use the Substitution Method to solve the following linear system. ( 3x − y = 5 x+y = 7

The Elimination Method 1. Add or subtract multiples of equations together so that one variable is eliminated. 2. Solve for remaining variable. 3. Plug the solution of one variable into any equation to solve for the other. 4. Write solution as a tuple (x1 , x2 , ..., xn ).

Example. Use the Elimination Method to solve the following linear system. ( 3x − y = 5 x+y = 7

What does Substitution/Elimination look like when the system is inconsistent (no solution) or dependent (infinitely many solutions)?

Example. Use Substitution or Elimination to solve the following linear system. ( 2x + 4y = −20 −3x − 6y = −36

Example. Use Substitution or Elimination to solve the following linear system. ( −4x + 9y = −27 8x − 18y = 54

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