Solving Systems of Equations Rachel Jumper http://t2.ftcdn.net/jpg/00/30/53/63/400_F_30536399_Ch4rzwGttuCQy73jCtyPIrGcxkmuGNaI.jpg

Hello! My name is Doug and I am going to be your guide for learning how to solve systems of equations! http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Buttons Click the addition sign to go to the next page. Click the subtraction sign to go to the previous page. Click on the house to go to the main menu page.

First thing’s first, are you a student or teacher?

Learning Environment The environment of the school is decent. It is a public school with a fair amount of resources. The class has access to a computer lab where there is a computer for each student.

Audience My students are Algebra I students in high school. They are mainly freshman. They have been learning about solving linear equations, so they are familiar with relating terms.

Objectives Given a system of linear equations, students will be able to tell how many solutions the system has and what that solution is with little to no error. Given a system of linear equations, students will be able to solve the system using one of the methods, elimination, substitution, or graphing, with little to no error.

Standards Al. 3. 1. Understand the relationship between a solution of a pair of linear equations in two variables and the graphs of the corresponding lines and solve pairs of linear equations in two variables by graphing, substitution, or elimination. Al. 3. 3. Solve problems that can be modeled using pairs of linear equations in two variables, interpret the solutions, and determine whether the solutions are reasonable.

Before getting into the methods of solving systems of linear equations, here is a little bit of basic information to get you started! http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

A system of linear equations can have: One solution Infinitely many solutions No solution

One Solution A system of linear equations with only one solution is where the equations only intersect at one point. http://www.jamesbrennan.org/algebra/systems/solution_set_files/image004.gif

Infinitely Many Solutions A system of linear equations has infinitely many solutions if the equations form the same line. (In other words, if the equations completely overlap.) http://mreaton.com/wp-content/uploads/2012/01/graphsysinfinite.jpg

No Solution A system of linear equations has no solution if the equations do not intersect. (Ex. Parallel lines) http://www.jamesbrennan.org/algebra/systems/solution_set_files/image010.gif

Great! Now that you know about the different types of solutions, it’s time to get started on solving the systems of equations!! http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Solving Systems of Linear Equations SOLVE BY GRAPHING Click on any of the lessons to start learning about solving systems of equations! SOLVE BY SUBSTITUTION SOLVE BY ELIMINATION http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg ASSESSMENT

Solving by Graphing ___ _____ -3y = -6x - 6 6x – 3y = -6 -6x -6x -3 -3 STEP 1: First, solve the equations for slope-intercept form, or in other words, solve each equation for y. ___ _____ -3y = -6x - 6 6x – 3y = -6 -6x -6x -3 -3 y = 2x + 2 4x + y = 8 -4x -4x y = -4x + 8

Please click here to review how to graph a line. Solving by Graphing STEP 2: After putting the equations in slope-intercept form, graph the lines. Please click here to review how to graph a line. y = -4x + 8 http://media1.shmoop.com/images/algebra-ii/alg2_ch2_narr_graphik_2.png y = 2x + 2

How to graph a line.

Solving by Graphing (1, 4) is the solution STEP 3: Find the point where the lines intersect. The point where the two lines intersect is the solution. (1, 4) is the solution http://media1.shmoop.com/images/algebra-ii/alg2_ch2_narr_graphik_2.png

GREAT! Now let’s practice by solving this system of equations: 4x – y = 4 and 4x + y = 4. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Practice: Solving by Graphing Solve for slope-intercept form. In order to solve for y, is the first step to subtract 4x from both sides or add 4x to both sides? 4x – y = 4 http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg -4x +4x

I’m sorry that is incorrect, please try again I’m sorry that is incorrect, please try again. Remember that in order to move a term, perform the opposite operation on the term. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

CORRECT! Great job! Now what is the next step? 4x – y = 4 -4x -4x -y = -4x + 4 Divide each side by -1 http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg Multiply each side by -1

I’m sorry that is incorrect, please try again I’m sorry that is incorrect, please try again. Remember that in order to remove a coefficient, perform the opposite operation on the term. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

CORRECT! Now that you solved 4x – y = 4 into slope-intercept form as y = 4x - 4, do the same for the equation 4x + y = 4. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

4x + y = 4 simplifies to… y = 4x + 4 y = -4x + 4 y = 4x - 4 y = -4x - 4

I’m sorry that is incorrect, please try again I’m sorry that is incorrect, please try again. Remember that in order to move a term, perform the opposite operation on the term. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Great! Now that you have solved the two equations for y, it is time to graph the lines. First, graph the line y = 4x – 4. Which graph is y = 4x – 4? A B Graphs: http://media1.shmoop.com/images/algebra-ii/alg2_ch2_narr_graphik_2.png Boy: http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

CORRECT. Now graph y = -4x +4 CORRECT! Now graph y = -4x +4. Which is the correct graph of y = -4x + 4? A B http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

I’m sorry, please try again I’m sorry, please try again. Remember that in the form y= mx + b, m is the slope and b is the y-intercept. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

CORRECT! Now where do the two lines intersect? (1, 1) (1,0) (-1, 0) http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg (0,1)

I’m sorry, please try again I’m sorry, please try again. Remember that the ordered pair is written as (x-value, y-value). http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

CORRECT! What type of solution did this system of equations have? Infinitely Many One solution http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg No solution

I’m sorry, please try again. Click here to review the different types of solutions to systems of equations. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Great job!! The system 4x – y = 4 and 4x + y = 4 has exactly one solution at (1,0). http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

A system of linear equations can have: One solution Infinitely many solutions No solution

One Solution A system of linear equations with only one solution is where the equations only intersect at one point. http://www.jamesbrennan.org/algebra/systems/solution_set_files/image004.gif

Infinitely Many Solutions A system of linear equations has infinitely many solutions if the equations form the same line. (In other words, if the equations completely overlap.) http://mreaton.com/wp-content/uploads/2012/01/graphsysinfinite.jpg

No Solution A system of linear equations has no solution if the equations do not intersect. (Ex. Parallel lines) http://www.jamesbrennan.org/algebra/systems/solution_set_files/image010.gif

Great. You have mastered solving systems of equations by graphing Great! You have mastered solving systems of equations by graphing! Click the home button to go back to the main menu!! http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Let’s solve x + 2y = 6 and 3x – 4y = 28! Let’s learn how to solve a system of equations using substitution. This process involves solving one equation for a variable, and then plugging that expression into the other equation. Let’s solve x + 2y = 6 and 3x – 4y = 28!

Solving with Substitution STEP 1: Solve at least one equation for one variable. x + 2y = 6 3x - 4y = 28 -2y -2y x = 6 – 2y

Solving with Substitution STEP 2: Substitute the resulting expression from STEP 1 into the other equation to replace the variable. Then solve the equation for the remaining variable. 3(6 – 2y) – 4y = 28 18 – 6y – 4y = 28 18 – 10y = 28 -18 -18 -10y = 10 -10 -10 y = -1 http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Solving with Substitution STEP 3: Substitute the value from STEP 2 into either equation and solve for the other variable. x + 2y = 6 3x – 4y = 28 x + 2(-1) = 6 x – 2 = 6 +2 +2 x = 6

Solving with Substitution STEP 4: Write the solution as an ordered pair. The solution to the system of equations of x + 2y = 6 and 3x – 4y = 28 is (6, -1).

Now let’s solve the system 2x + y = 3 and 4x + 4y = 8 using substitution. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Solve 2x + y = 3 for the variable y. Solve for a variable… Hint: For this equation, solving for y would be easier than solving for x because the coefficient in front of y is one. Solve 2x + y = 3 for the variable y. y = 3 – 2x y = 3 + 2x

Correct! Now click on the red box to substitute (3 – 2x) in for y in the other equation which was 4x + 4y =8. y = 3 – 2x 4x + 4 y = 8

I’m sorry, that is incorrect. Please try again I’m sorry, that is incorrect. Please try again. Remember when you are moving a term to another side, perform the opposite operation on the term. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Simplify: 4x + 4 (3 – 2x) = 8 1 4x + 12 – 8x = 8 2 4x + 12 – 6x = 8

Find x: 12 – 4x = 8 x = 2 x = -1 x = 1 x = -2

Great, x does equal 1! Now click on the blue box to substitute 1 in for x in the other equation which was 4x + 4y =8. x = 1 4 x + 4y = 8

I’m sorry, that is not correct. Please try again. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Find y: 4 + 4y = 8 y = 2 y = 1 y = -1 y = -2

Correct! Great job! y = 1 http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

I’m sorry, that is incorrect. Please try again. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

2x + y = 3 and 4x + 4y = 8 has a solution at… (1,1) (1,-1) (-1,1) (-1,-1)

I’m sorry, that is incorrect. Please try again I’m sorry, that is incorrect. Please try again. Remember that a solution is written as (x, y). http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Correct! The solution is (1, 1). http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Great. You have mastered solving systems of equations by substitution Great! You have mastered solving systems of equations by substitution! Click the home button to go back to the main menu!! http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Solving by Elimination STEP 1: Write the system so like terms with the same or opposite coefficients are aligned. 4x + 6y = 32 3x – 6y = 3

Solving by Elimination STEP 2: Add or subtract the equations, eliminating one variable. Then solve the equation. 4x + 6y = 32 3x – 6y = 3 7x + 0 = 35 (+) 7 7 x = 5

Solving by Elimination STEP 3: Substitute the value from step 2 into one of the equations and solve for the other variable. Write the solution as an ordered pair. 3x – 6y = 3 3(5) – 6y = 3 15 – 6y = 3 -6y = 3 y = -1/2

4x + 6y = 32 and 3x – 6y = 3 have exactly one solution at (5, -1/2). http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Great! Now solve 2x – y = 4 and 7x + 3y = 27 using elimination. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

First, eliminate the y-term, and solve for x. Click here for a hint. First, eliminate the y-term, and solve for x. 2x – y = 4 7x + 3y = 27 x = 6 x = 1 http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg x = -3 x = 3

Solving with Elimination In order to be able to eliminate a term, the coefficients must either be the same or opposites. 2x – y = 4 and 7x + 3y = 27 does not currently have the same or opposite coefficients. In order to create this, you can multiply one of the equations by a constant. In this case, multiplying (2x – y = 4) by 3 would create opposite coefficients for the y-term. 3(2x – y = 4) 6x – 3y = 12

Solving with Elimination Now just eliminate the y-term by adding the two equations together. 6x – 3y = 12 7x + 3y = 27 13x = 39 (+)

Correct! x = 3. Now solve for y. 2x – y = 4 7x + 3y = 27 y = 1 y = 2 y = -2 y = 3

I’m sorry. Please try again. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

I’m sorry. Please try again. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Correct! y = 2. So the solution to 2x – y = 4 and 7x + 3y = 27 is… (1, 1) (1, 2) (2, 3) (3, 2) http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

I’m sorry. Please try again. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Correct! The solution to 2x – y = 4 and 7x + 3y = 27 is (3, 2). http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Great. You have mastered solving systems of equations by elimination Great! You have mastered solving systems of equations by elimination! Click the home button to go back to the main menu!! http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

ASSESSMENT BEWARE: Once you push the plus sign to enter the assessment, you cannot return back to the main menu or lessons. GOOD LUCK! http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

Please write your answers on a piece of paper Please write your answers on a piece of paper. You are to turn your paper in when you have finished. Please show all of your work. http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg

How many solutions? ONE SOLUTION INFINITELY MANY SOLUTIONS NO SOLUTION 1. How many solutions does 2x – 8y = 6 and x – 4y = 3 have? ONE SOLUTION INFINITELY MANY SOLUTIONS NO SOLUTION

What’s the solution? 2. Use graphing, substitution, or elimination to solve: y = 5x + 1 4x + y = 10 (1, 6) (6, 1) (2, 1) No solution

What’s the solution? 3. Use graphing, substitution, or elimination to solve: 8x + 5y = 38 -8x + 2y = 4 (6, 1) (-6, 1) (1, 6) No solution

What’s the solution? 4. Use graphing, substitution, or elimination to solve: 2x – 3y = -9 -x + 3y = 6 (3, 3) (-3, 1) (-3, 3) (1, -3)

What’s the solution? 5. Use graphing, substitution, or elimination to solve: 12x – 3y = -3 6x + y = 1 (1, 0) (0, -1) (-1, 0) (0, 1)

You are now finished! Please turn in your paper! Great job! http://famousjojo.com/wp-content/uploads/2010/01/Mamas-Boy-Nerd-Cartoon.jpg