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**Solving Systems of Equations**

Rachel Jumper

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Hello! My name is Doug and I am going to be your guide for learning how to solve systems of equations!

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**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.

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

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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.

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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.

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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.

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Standards Al 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 Solve problems that can be modeled using pairs of linear equations in two variables, interpret the solutions, and determine whether the solutions are reasonable.

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Before getting into the methods of solving systems of linear equations, here is a little bit of basic information to get you started!

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

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One Solution A system of linear equations with only one solution is where the equations only intersect at one point.

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**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.)

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No Solution A system of linear equations has no solution if the equations do not intersect. (Ex. Parallel lines)

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Great! Now that you know about the different types of solutions, it’s time to get started on solving the systems of equations!!

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**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 ASSESSMENT

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**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 x y = 2x + 2 4x + y = 8 -4x x y = -4x + 8

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**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 y = 2x + 2

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How to graph a line.

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**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

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**GREAT! Now let’s practice by solving this system of equations:**

4x – y = 4 and 4x + y = 4.

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**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 -4x +4x

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**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.

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**CORRECT! Great job! Now what is the next step?**

4x – y = 4 -4x x -y = -4x + 4 Divide each side by -1 Multiply each side by -1

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**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.

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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.

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

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**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.

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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: Boy:

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**CORRECT. Now graph y = -4x +4**

CORRECT! Now graph y = -4x Which is the correct graph of y = -4x + 4? A B

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**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.

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**CORRECT! Now where do the two lines intersect?**

(1, 1) (1,0) (-1, 0) (0,1)

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**I’m sorry, please try again**

I’m sorry, please try again. Remember that the ordered pair is written as (x-value, y-value).

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**CORRECT! What type of solution did this system of equations have?**

Infinitely Many One solution No solution

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**I’m sorry, please try again.**

Click here to review the different types of solutions to systems of equations.

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**Great job!! The system 4x – y = 4 and 4x + y = 4 has exactly one solution at (1,0).**

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

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One Solution A system of linear equations with only one solution is where the equations only intersect at one point.

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**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.)

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No Solution A system of linear equations has no solution if the equations do not intersect. (Ex. Parallel lines)

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**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!!

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**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!

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**Solving with Substitution**

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

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**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 -10y = 10 y = -1

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**Solving with Substitution**

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

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**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).

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**Now let’s solve the system 2x + y = 3 and 4x + 4y = 8 using substitution.**

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**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

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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

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**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.

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

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

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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

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**I’m sorry, that is not correct. Please try again.**

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

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Correct! Great job! y = 1

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**I’m sorry, that is incorrect. Please try again.**

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**2x + y = 3 and 4x + 4y = 8 has a solution at…**

(1,1) (1,-1) (-1,1) (-1,-1)

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**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).

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**Correct! The solution is (1, 1).**

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**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!!

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**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

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**Solving by Elimination**

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

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**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

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**4x + 6y = 32 and 3x – 6y = 3 have exactly one solution at (5, -1/2).**

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**Great! Now solve 2x – y = 4 and 7x + 3y = 27 using elimination.**

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**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 x = -3 x = 3

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**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

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**Solving with Elimination**

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

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**Correct! x = 3. Now solve for y.**

2x – y = x + 3y = 27 y = 1 y = 2 y = -2 y = 3

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**I’m sorry. Please try again.**

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**I’m sorry. Please try again.**

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**Correct! y = 2. So the solution to 2x – y = 4 and 7x + 3y = 27 is…**

(1, 1) (1, 2) (2, 3) (3, 2)

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**I’m sorry. Please try again.**

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**Correct! The solution to 2x – y = 4 and 7x + 3y = 27 is (3, 2).**

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**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!!

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ASSESSMENT BEWARE: Once you push the plus sign to enter the assessment, you cannot return back to the main menu or lessons. GOOD LUCK!

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**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.

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**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

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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

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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

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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)

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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)

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**You are now finished! Please turn in your paper! Great job!**

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