Unit 5 Linear Systems

Lesson 1 Solutions of a system

Terms Definition Diagram

consistent equations

a system of linear equations that contain at least one common point

dependent equations

a system of linear equations that rely on each other for the algebraic or graphic form of the equation

equivalent equations

equations having all common solutions

inconsistent equations

a system of linear equations that do not contain any common points

independent equations

a system of linear equations that do not rely on each other for the algebraic or graphic form of the equation

infinitely many solutions

A set of linear equations that coincide and share every point as a point of intersection. Also known as a dependent and consistent solution.

no solutiona set of parallel lines that will never share a point of intersection. Considered to be an inconsistent solution ("empty set"-“{}”)

one solution

a set of linear equations that share a common point known as the point of intersection (x,y). The solution, (x,y) is an independent and consistent solution

There are three possible solution types to a system of equations.

Graphically, a pair of consistent equations shows one distinct intersection, or one solution.

A pair of parallel lines shows no solution as they will never intersect because they continue to increase or decrease at an equal rate of change.

In the case that a system of equations graphs a pair of coinciding, or equivalent equations, there are an infinite number of solutions.

In the same way, there are three possible solution types to a system of linear equations when the system is solved algebraically.

One SolutionA system of equations has one solution when the solution provides one distinct intersection point, or a specific x- and y-value. Only these specific x- and y-values will make the system true. To determine if there is one distinct intersection point between a pair of linear equations, rewrite each equation in slope-intercept form. If the equations do not have an equal slope, then we can predict that there is one distinct solution to the system.

Example:Use algebraic rules of equations to predict the solution type to the system of equations.

Solution:

Solve for the slope-intercept form of each equation.

x + y = 20x - x + y = 20 - xy = -x + 20

  

2x + y = 362x - 2x + y = 36 - 2xy = -2x + 36

As you can see, the equations do not have an equal slope. Therefore, we can predict that there will be one distinct solution (x, y) to the system of linear equations.

No SolutionWhen a system of equations produces a pair of parallel lines, there will be no solution to the system. In other words, the equation has the same exact slope but a different y-intercept, therefore they will never intersect.

Example:Use algebraic rules of equations to predict the solution type to the system of equations.

Solution:

Use algebraic rules of equation solving to rewrite each equation in slope-intercept form. Compare the slopes and y-intercepts of the equations.

y = 30 - 3x 9x + 3y = 46

y = -3x + 30

9x - 9x + 3y = 46 - 9x

y = -3x + 

slope = -3 slope = -3

y-intercept = 30 y-intercept = 

The two equations have the same slope but they do not share the same y-intercept. Therefore, we can predict that there will be no solution to the system of equations because they are parallel lines and will never intersect.

Infinitely Many SolutionsA pair of dependent lines has a corresponding system containing equivalent equations. Remember that when the lines are dependent, they are coinciding and have an infinite number of solutions. This means that all (x, y) values of the lines will make the system true.

Example:Use algebraic rules of equations to predict the solution type to the system of equations.

Solution:

Use algebraic rules of equation solving to rewrite each equation in slope-intercept form. Compare the slopes and y-intercepts of the equations.

y = 10 - 2x 6x + 3y = 30

y = -2x + 10

6x - 6x + 3y = 30 - 6x

y = -2x + 10

slope = -2 slope = -2

y-intercept = 10 y-intercept = 10

The two equations have equal slopes and y-intercepts. Therefore, we can predict that there will be an infinite number of solution points that will satisfy the system of equations.

Lesson 2 Graphing Systems of Equations

To solve a system of linear inequalities, you'll need to graph each inequality in the same coordinate plane. This means that you will need to repeat the above steps for each inequality. When you are finished, you'll look for the intersection (overlap) of the two solution sets.)

Lesson 5 Substitution MethodHere is a summary of the steps we used to solve the above system:

1. Substitute an equivalent expression for a variable into one of the equations. (We substituted an expression for y.)

2. Solve the resulting equation for the other variable. (We solved for x.)3. Substitute that value into the one of the original equations. (We put the value of x

into one of the equations.)4. Solve for the other variable. (We solved for y.)5. Check the ordered pair in the other equation.6. State the solution set.

Example:Solve the following system by the substitution method.

2x + y = 2x - y = 1

Solution:Solve 2x + y = 2 for y because it has a Solve x - y = 1 for x because it has a

coefficient of 1:y = 2 - 2x

coefficient of 1:x = y + 1

Substitute 2 - 2x into the other equation in place of y:x - (2 - 2x) = 1

Substitute y + 1 into the other equation in place of x:2(y + 1) + y = 2

Solve the equation for x:x - (2 - 2x) = 1x - 2 + 2x = 13x - 2 = 13x = 3x = 1

Solve the equation for y:2(y + 1) + y = 22y + 2 + y = 23y + 2 = 23y = 0y = 0

Substitute the value of x into one of the original equations and solve for y:x - y = 11 - y = 1y = 0

Substitute the value of y into one of the original equations and solve for x:x - y = 1x - 0 = 1x = 1

Check the solution (1, 0) in the other equation:2x + y = 22(1) + 0   22 = 2 (True)

Check the solution (1, 0) in the other equation:2x + y = 22(1) + 0   22 = 2 (True)

The solution set of the system is {(1, 0)}.

To solve a linear system by substitution:

1. Solve one of the equations for a variable.2. Substitute the equivalent expression for the variable in step 1 into the other

equation.3. Solve the resulting equation for the other variable.4. Substitute that value into the one of the original equations.5. Solve for the other variable.6. Check the ordered pair in the other equation.7. State the solution set.

Substitution: Comparison Method

It is important to remind ourselves that linear systems may also have zero or an infinite number of solutions. If the lines are parallel (inconsistent system), then there is no solution. If the lines are the same line (equivalent system), then the solution set is the equation.

Lesson 6 Addition Solution

Example:Solve the system by elimination.

x + y = 8x + y = 6

   

   There is no solution.The lines are parallel, and the system is inconsistent.

Example:

The solution set is infinite.The equations graph the same line, and the system is equivalent.

Example:

   

   

Example:

 

The solution is (5, -5).

   

Lesson 7 Matrices

determinant A single number obtained from a matrix that reveals some aspects of a matrix's properties

coefficient the constant preceding the variables in a product

matrix a rectangular array made up of rows and columns

Lesson Outline:

A matrix is a rectangular array of numbers. The determinant of a 2 x 2 matrix is the difference of the cross products: (row 1,

column 1)(row 2, column 2) - (row 1, column 2)(row 2, column 1). The system, x, and y determinants can be used to solve a system of two linear

equations. To solve a linear system using determinants, the equations must be in

determinant form.

MatricesA matrix is a rectangular table of numbers. A matrix consists of rows and columns. Look at the following matrix having two rows and two columns.

Solving systems of equations

Term Definition Example

standard formthe form Ax + By = C of a linear equation, where A, B and C are integers

determinant forma system of linear equations in which each equation is written in standard form

system determinant

the determinant found when column 1 consists of the x-coefficients and column 2 consists of the y-coefficients of a linear system

x-determinant

the determinant found when column 1 consists of the constants and column 2 consists of the y-coefficients of a linear system

y-determinant

the determinant found when column 1 consists of the x-coefficients and column 2 consists of the constants of a linear system

Determinants provide a useful tool for solving problems. Determinants use the relationship between the x and y coefficients and the constants of equations to identify a solution set.

The determinant form of a system of linear equations is one in which the equations are written in standard form, Ax + By = C , where A, B, and C are integers . Recall that A and B are the coefficients on the variables x and y, and C is the constant. We use these coefficients to form the determinant matrix. The determinant matrix is shown for the system of equations below:System of equations:Qx + Ry = F Sx + T y = G Determinant Matrix:

 Note that for the system determinant the x-coefficients form column 1 and the y-coefficients form column 2.

The matrix for the x-determinant is formed by replacing the x-coordinates in the system matrix with the constants of the system of equations:

The matrix for the y-determinant is formed by replacing the y-coordinates in the system matrix with the constants of the system of equations:

Example:

Lesson 8 Fractional Coefficients

When an equation involves fractional coefficients, we may rewrite the equation in order to eliminate the fractional values. This process is often referred to as "clearing fractions."

Example:Solve the system of equations.

y =  x - 62x + y = -2

Solution:First, use the multiplication property of equality to clear the fraction from the first equation:

3[y =  x - 6]

3y = 3( x) - 3(6)3y = 2x - 18Now write the system in standard form:-2x + 3y = -18

2x + y = -2Use an algebraic method to solve:

 we used Cramer's rule to calculate the solution to the system. It is important to note that, after you have cleared fractions, any method for solving a system of equations (graphing, substitution, elimination, or matrices) can be used. 

Example:

Solution:Key point!We chose to use the elimination method here because the coefficients on y are additive inverses (opposites).