## 10.8.3 Number of solutions to a system.

Consider the system of equations:

$3x + 2y = 16$

$y = -\frac32 x + 8$

One solution for both of these equations is at (2,5).  However, there is another solution at (0,8).  Since the solution is represented by the point of intersection, how is it possible for 2 lines to intersect at more than 1 place?

Resources:

• Notes: Number of solutions

Assignment

• p455 #1-3, 5-7, 9, 12-17 *18-20

Things you should know after today:

• how to determine the number of solutions given the graphs of a system
• the relationship between number of solutions and the slopes and y-intercepts of the lines in a system
• how to determine the number of solutions by inspection of the system

## 10.8.2 Modeling Systems of Equations

Many real world problems can be represented as linear equations, and the challenge lies in determining the equations first.  Once the equations have been determined, then you can easily find the solution to the sytem graphically using technology (eg Desmos) or by graphing with pencil and paper and estimating the point of intersection (the solution) for the system.

You will need to be able to determine what the variables will represent, and then create two equations, each relating both variables.

Resources

• Notes
• Blank Assignment Sheet

Assignment

• p440 #1-9 11 13-15 17 19 20
• Solving a Problem using a System of Equations

Things you should be able to do after today:

• create a linear system from a word problem
• explain what the coordinate that is the point of intersection represents in terms of the word problem

## 10.8.1 Systems of Linear Equations

A linear equation involving two variables has an infinite number of solutions, all of which lie on the graph which makes a diagonal line.  A second linear equation will also have its own infinite number of solutions.

When the two equations are combined into a system, there may be 1 coordinate that is common to both of the systems.  This is located at the point of intersection of the two lines, and represents the solution to the entire system.

Once you find the solution, you can easily test whether it is the solution to the system by substituting the x and y values into both of the equations in the system.  If it really is the solution, then it will make both equations true.

Resources

• Notes

Assignment

• p424 #1-4, #5ac #6, #7ac, #8-18 20 *23

Things you should be able to do after today:

• determine the solution for a system of linear equations graphically
• relate the coordinate for the solution to a linear system to the equations in the system

## 10.9.1 Solving Systems by Substitution

One of the algebraic methods for solving systems of equations is to solve by substitution.  It is based on the idea that if a = b, and b = 25, then you can also say that a = 25.

Sounds simple, right?  Imagine a more complicated scenario:

therefore

And then the equation can be solved for x.

Resources:

• Notes

Assignment

• p475 #1, 3, 4-9, 11, 13, 14, 17-20, 25 *21 22 24

Things you should be able to do after today:

• solve a system of equations by substitution
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