Solving Systems of Equations using Elimination

Topics for Today:

We are still working on systems of linear equations.  Today, we discussed elimination method, and with a system of two equations, this method is really the preferred one.

Steps for Solving Using Elimination Method

1. In your original system, make sure both equations are in the same form (standard form works best!).  Line your equations up so the variables are aligned in columns.  
2. Determine which variable should be eliminated.  Look for matching numbers and opposite signs or create them using multiplication.  You may have to multiply both equations so that you can eliminate one variable.  
3. Eliminate the chosen variable.  Solve for the other variable.  
4. Take the value you found in Step 3 and substitute it into one of the original equations to solve for the other variable. 
5. Identify your solution – it will be an ordered pair!
6. Check both original equations with the solution you found.  

Vocabulary:  elimination method

Sections Covered in Textbook:

7-3: Solving Systems Using Elimination (pages 353-359)

Resources & Tutorials:

1. How do you solve a system of equations using the elimination by addition method? 
2. How do you solve a system of equations using the elimination by multiplication method?
3. What's another way of solving a system of equations using the elimination by multiplication method?

Solving Systems of Equations Using Substitution

Topics for Today:

We are still working on solving systems of equations and introduced a new method today that uses algebra instead of graphing.  We have discussed the limitations of the graphing method, and why we might use algebra instead.  There are three algebraic methods used to solve systems:  substitution method, elimination method, and the matrix method.  We will learn and practice the first two; solving of matrices with systems of 2 or more equations is covered in Algebra II.

The general process for solving systems algebraically is the same.  First, we solve for one of the variables; then we substitute that solution into one of our equations to find the second variable.

For the substitution method, we follow this process:

1. In the original system, see if one variable is isolated; if not, then isolate a variable. 
2. Substitute the expression into the second equation.
3. Solve the equation for the first variable.
4. Substitute the solution found in step 3 into one of the original equations to solve for the other variable. 
5. Identify the solution as an ordered pair.
6. Check both original equations to ensure the solution works for both. 

Vocabulary:  substitution method

Sections Covered in Textbook:

7-2: Solving Systems Using Substitution (pages 347-351)

Resources & Tutorials:

1) How to solve a system using substitution method.
2) Solving Systems of Equations by Substitution.

Solving Systems of Equations by Graphing

Topics for Today:

We are still working with graphing linear equations, but we've expanded our conversation to include systems of linear equations.  A linear system of equations is simply two or more linear equations containing the same variables.  When we deal in generic equations, we almost always use the variables x and y; however, when we use systems to solve real problems, we may define our variable with different letters that better match our problem.  For instance, if we are talking about costs and revenue, we may choose to use c and r for our variables.

Systems of two linear equations have three possible types of solutions because they are based upon where two lines intersect on a plane:  they either intersect nowhere, intersect at one point, or intersect at every point.  If there is a solution, it is represented as an ordered pair.

Summary of Systems of Equations
(Click Graphic to Enlarge)

Vocabulary:  system of linear equations, solution of a system of linear equations

Sections Covered in Textbook:

7-1: Solving Systems by Graphing (pages 340-345)

Resources & Tutorials:

1) What is a system of linear equations?
2) How do you solve a system using graphing?
3) What is a solution to a system of equations? 
4) What are the three types of solutions to a system of equations?

Parallel and Perpendicular Lines

Topics for Today:

The slope of two lines can produce a special relationship between those lines.  Two such relationship are parallel lines and perpendicular lines.  Parallel lines exist in the same plane but will never intersect, and they always have the same slope.  Perpendicular lines are special because when they intersect, the lines form 90° angles.  The slopes of perpendicular lines are negative reciprocals of each other, and when those slopes are multiplied together, the result is -1.

We will be analyzing the slopes of two lines to determine if either relationship exists, and we will be deducing linear equations from a given point that is either parallel or perpendicular to the given line.

Parallel and perpendicular lines are always determined by the relationship of their slopes!

Vocabulary: parallel lines, perpendicular lines, negative reciprocal

Sections Covered in Textbook:

6-5: Parallel and Perpendicular Lines (pages 311 - 316)

Resources & Tutorials:

1) How do you find the slope of a line if you have a parallel line?
2)  How do you write an equation of a line in slope-intercept form if you have one point and a parallel line?
3) How do you find the slope of a line if you have a perpendicular line?
4) How do you write an equation of a line in slope-intercept form if you have one point and a perpendicular line?
5) How to tell if lines are parallel, perpendicular, or neither.

Absolute Value Translations

Topics for Today:

Today we began our discussion of graphing absolute value functions through a DESMOS activity in class.  The purpose of the activity was to allow students to explore the various parts of an absolute value equation and to draw their own conclusions about how changing various pieces of the equation affects how the graph looks.  All of these variations are called translations.

Vocabulary: absolute value graph, translation, vertex

Sections Covered in Textbook:

6-7: Graphing Absolute Value Equations (pages 325-329)

Resources & Tutorials:

1)  Graphing Absolute Value Equations Introduction
2)  Shifting Absolute Value Graphs
3) Amplify Absolute Value Activity - See Google Classroom for link.

Point-Slope Form

Topics for Today:

Today we discussed the last line form, point-slope form, through a PowerPoint presentation (link below).  

The three different forms of a linear equation are all useful for different reasons.  Each one has a pattern to follow.  Once the patterns are mastered, linear graphing becomes much easier.  

Vocabulary: point-slope form

Sections Covered in Textbook:

6-4: Point-Slope Form and Writing Linear Equations (pages 304-309)

Resources & Tutorials:

1) What is the point-slope form of a linear equation?

2) How do you write an equation of a line in point-slope form if you have the slope and one point?

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3) Exploring Point-Slope Form (PowerPoint)

Standard Form

Topics for Today:

Our discussion about linear equations continued today.  We have already explored slope and slope-intercept form of a line.  Today, we looked at a different form - standard form.  The standard form of a line is defined as a linear equation such that

Ax + By = C

A, B, and C must all be integers.

A must be positive.

Although it's easy to visualize a line that is in slope-intercept form (the form we worked with yesterday), it's very easy to find both the x- and y-intercepts when a line is in standard form.  These intercepts are where the line crosses the x- and y-axes, when one of our coordinates is zero.  Solving the equation when substituting a zero for a value is a quick process, because multiplying by zero removes the variable from the equation.  Once we find our intercepts, it's very easy to graph our equation.

Standard form can be nice for students who are not fond of working with fractions, and we'll be using standard form when we move to solving systems of equations in the next chapter.  In addition, many of the other graphs that students will see in later mathematics classes are written in standard form with the variables all on one side of the equation.  Comfort with standard form will help students cope when they are introduced to more complicated equations.

Vocabulary: standard form of a line, x-intercept

Sections Covered in Textbook:

6-3: Standard Form (pages 298-302)

Resources & Tutorials:

1) What is the standard form of a linear equation?
2) How do you use x- and y-intercepts to graph a line in standard form?