Solve Linear Systems Algebraically Part I

Chapter 3.2

Solutions of Linear Systems of Equations

• A linear system of equations will always have one of the following as a solution• Exactly one solution in x and y (the lines intersect in a single point)• An infinite number of solutions (the lines coincide and share all points)• No solution (the lines are parallel and never intersect)

• The next slide shows how graphs of the last two would look

Solutions of Linear Systems of Equations

Solve Linear Systems Algebraically

• Although it is possible to solve a linear system of equations by graphing, this is seldom the best method

• The reason is that, if the solution is not an ordered pair with integer coordinates, then the point of intersection has fractional values

• These are usually impossible to read unless the coordinate plane is broken in the right fractional values

• The best method for solving a linear system of equations is by algebraic methods

Solve Linear Systems Algebraically

• You will learn about two such methods

• The first is called the substitution method

• The second is called the elimination method (or sometimes it is called the addition method)

• In today’s lesson you will solve linear systems by the substitution method

• This method is best used when one or both equations are solved for either y or for x

The Substitution Method

• Suppose you are to solve a linear system of equations like the one below

• Since the solution is the point that is common to both lines, then the x and y values from the first equation must be the same as the x and y values from the second equation

• This means that we can substitute the right part of the first equation into y in the second equation

The Substitution Method

Substitute this:

here:

and solve for x

The Substitution Method

So we have part of the solution:

We need to find y to complete the solution. Do this by substituting for x in the first equation.

The Substitution Method

The solution is

The Substitution Method

• Some linear systems might have both equations solved for y, like the one shown below

• The substitution method is the same: replace y in either equation with the right side of the other equation

The Substitution Method

Substitute this:

here:

and solve for x.

The Substitution Method

So we have part of the solution:

To find y, substitute this value into either equation.

The Substitution Method

The solution is .

A System With No Solution

• How would you know when a system of linear equations has no solution?

• The following example shows what to look for

Use the substitution method

A System With No Solution

• How would you know when a system of linear equations has no solution?

• The following example shows what to look for

• You should get something like , or possibly some other equation that is false

• When this happens, you conclude that the system has no solution

A System With Infinite Solutions

• How do you know when a system has an infinite number of solutions?

• The next example illustrates

• Use the substitution method

A System With Infinite Solutions

• How do you know when a system has an infinite number of solutions?

• The next example illustrates

• You should get something like or possibly

• Both of these are always true, so the system has an infinite number of solutions

Guided Practice

Solve the following systems of linear equations by the substitution method.

1.

2.

3.

4.

Exercise 3.2a

• Handout