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?Math 1??Take Home Packet #3????School: _____________________________??Teacher: __________________________??Name: _______________________________??Date: _________________????Table of Contents?April 23rd?Topic:?Graphing Systems and Solving Systems by Substitution??April 24th?Topic:?Solving Systems by Elimination??April 27th?Topic:?Solving Systems using Desmos??April 28th?Topic:?Solving Systems Word Problems??April 29th?Topic:?Graphing Linear Inequalities??April 30th?Topic:?Graphing Systems of Linear Inequalities??May 1st?Topic:?Exponent Rules and Operations with Polynomials??May 4th?Topic:?Factoring the GCF??May 5th?Topic:?Factoring Trinomials (a =?1)??May 6th?Topic:?Factoring Trinomials (a > 1)??May 7th?Topic:?Factoring Special Cases and Mixed Factoring??May 8th?Topic:?Graphing Quadratics and Parts of a Parabola??May 11th?Topic:?Solving Quadratics by Factoring??May 12th?Topic:?Solving Quadratics by Factoring Special Cases and?Sq?Roots??May 13th?Topic:?Word Problems with Quadratics??May 14th?Topic:?Word Problems with Quadratics??May 15th?Topic:?Solving Quadratics Mixed Review??April 23rd Notes 1: Identify Solutions to a System What is a system? In solving a system of equations, we try to find values for each of the variables that will satisfy every equation in the system.A solution to a system of equations in two variables is ONE point that is a solution or point of intersection.Example: Is this point the solution to the system given?Given Point: (2, -2)System:y=-12x+1y=-2x-2Solution: Plug in the points to see if they satisfy both equations-2=-122+1-2= -1+1-2≠0The point is NOT a solution to the system.YOU TRY! Determine whether the ordered pair is the solution to the system of equations given. y=-2x-1y=x-4 (-3, 1) y=7x-3y=x+3 (1, 4)y=8x 6x-y=12 (2, 16)4x-10y=63x-20y=17 (-1, -1)y=8x 6x-y=12 (2, 16)April 23rd Notes 2: Graphing a System Step 1: Write each equation in slope intercept form (y = mx +b).Step 2: Graph both of the equations on the same coordinate plane.Step 3: Determine the point of intersection which is your SOLUTION!Example: y=4x-4y=x+2Now you try! Graph on the provided graph paper. 1. 2. 3. 4. 5. 2x-8=4y -x=-2y-4April 23rd Notes 3: Solving a System by Substitution If you know the value of one variable in the system of equations, you can find the solution for the system by substituting the known value of the variable into the other equation. *Note* If you do NOT know the value of one variable, then you should solve one equation for one of the variables.Example: -8x+2-2x=12 Since we know y=-2x, then we substitute -2x in place of y. -8x-4x=12 Now, we solve for x!-12x=12x=-1 Now that we know x=-1, we will substitute that in place of x in the second eqn.y=-2-1=2 The solution is the point (-1, 2).Now you try! 1. 2. 3. 4. 5. 6. April 24th Notes: Solving a System by EliminationStep 1: Use the Addition Property of Equality to combine the equations and to eliminate one of the variables.Step 2: Solve the resulting equation for the remaining variableStep 3: Substitute the value of the variable back into one of the equations to determine the value of the other variable.Example:2x+y=-10-2x+3y=-64y= -16ADD the two equations together, 2x and -2x eliminate!y= -4Solve for y!2x+-4= -10Substitute -4 in place of y in one equation.2x-4=-10ADD 4 to both sides2x= -6DIVIDE by 2x=-3The solution is (-3, -4)Now you try!8x+4y=-24-2x+4y=26 -9x-5y=-7 6x+7y=234x+4y=0-10x-5y=-25 8x-6y=-4 -4x+3y=2 18x+20y=-12-9x-10y=158x+7y=-11-8x-8y=16x+10y=15-x-9y=-13-9x+5y=-239x-5y=232x+10y=6-2x+4y=8April 27th Notes: Solving Systems using DesmosFinding the solution of the system of equations y=12x-2y=14x+9Step 1: Type the first equation into the first line and second equation into the second line in Desmos.Step 2: Move the graph by using your cursor and dragging the screen (if necessary) so that you can see the point of intersection on your screen.Step 3: Click on the point of intersection. That is your solutionFind the solution to each system of equations. Use y=-23x+502x-3y=67x-3y=9 18x+20y=-12 -9x-10y=15 Practice Solving Systems with a Graphing Calculator4. 5.6. y=52x+1y=2x-37. 8. 9. Choose the best method for solving the following systems. April 28th Notes: Solving Systems of Equations WORD PROBLEMS1- Determine the information that you know, and what you are attempting to find.2- Assign a VARIABLE to each UNKNOWN!3- Write equations to represent the situations described.4- Choose the best method to solve the system.5- Solve the system. Check to make certain your answer makes sense. You Try!A 180-ft length of rope must be cut into two pieces. One piece has to be three times longer than the other. How long will each piece be?For an experiment, 40 volunteers have to be divided into two groups. One group must have 4 more than 5 times the number of people in the other group. How many people should be in each group?In an algebra class of 28 students, the number of girls is 5 less than twice the number of boys. How many boys and how many girls are in the class?A basketball center made 23 baskets for a total score of 34. How many field goals did the center make? (Hint: A field goal is worth 2 pts and a free throw is worth 1 pt.)The sum of two numbers is 12 and their difference is 4. Find the numbers.Ling Su is 5 years older than Betty. The sum of their ages is 27. How old are Ling Su and Betty.The sum of a number and twice another number is 16. The second number is 4 less than the first. Find the numbers.Three times Ravi’s age is 6 more than twice Anna’s age. The sum of their ages is 32. How old are Ravi and Anna?April 29th Notes: Graphing Linear InequalitiesStep 1: Graph the boundary line using the strategies learned for graphing linear equations in earlier units.For ≥and ≤ , use a SOLID LINEFor >and <, use a BROKEN LINE.Step 2: Test points to determine which side of the boundary line contains the solutions for the inequality.Step 3: Shade the side of the boundary that contains the solution.Examples: Now you try! April 30th Notes: Graphing Systems of Linear InequalitiesStep 1: Graph both Inequalities on the same grid (Using different patterns or colors is helpful to highlight the overlap)Step 2: The solution to the system are all points included in the intersection. Examples: Now you try: May 1st Notes 1: Exponent RulesMay 1st Notes 2: Operations with PolynomialsMay 4th Notes: Factoring the GCFMay 5th Notes: Factoring trinomials with a = 1. May 6th Notes: Factoring trinomials with a > 1. Now you try! May 7th Notes: Factoring special cases and MIXED factoring. Now you try! May 8th Notes: Graphing a Quadratic and Parts of a Parabola Parts of a Parabola PRACTICE: Domain and Range:Now you try!May 11th Notes: Solving a Quadratic Function by Factoring May 12th Notes: Solve Quadratics by Factoring special cases and square rootSolve by factoring a special case. (See notes on difference of squares and perfect square trinomials.)x2-16=04x2-25=0x2+6x+9=0x2-10x+25=0x2-100=02x2+28x+98=02x2-10x+50=0May 13th Notes: Solving Quadratic Word Problems by FactoringNotes: Identify key words to determine whether you need to find the Vertex (max/min) or factor.Key words that let you know to factor: Hit the ground, horizontal distance, how long has the object been in the air, review notes on May 11th to recall how to factor to solve. Key words that let you know to find the max/min: How high, Vertical height, hit a maximum height, minimum profit, review notes on May 8th to recall how to find the vertexSolve the following by factoring.Matthew throws a baseball into the air. The path of the baseball is modeled by the following function ht= -16t2+64t When does the ball hit the ground?At what time does the ball hit a maximum height?What is the maximum height?A frog’s jump is modeled by the function ht= -5t2+30t. What is the horizontal distance the frog travels in any jump?What are the values of x in the following quadratic? 2x2+5x-3Susie launches a rocket in the air and the path of the rocket is modeled by the following function ht=-16t2+96t+16After how many seconds, does the rocket hit the ground?At what time does the rocket hit a maximum height?What is the maximum height?A model rocket is launched from the roof of a building. It’s flight path is modeled by the following function.hx=-5x2+5x+30When does the rocket hit the ground?At what time does the rocket hit a maximum height?What is the maximum height?May 14th Notes: Solving Quadratic Word Problems by FactoringNotes: Identify key words to determine whether you need to find the Vertex (max/min) or factor.Key words that let you know to factor: Hit the ground, horizontal distance, how long has the object been in the air, review notes on May 11th to recall how to factor to solve. Key words that let you know to find the max/min: How high, Vertical height, hit a maximum height, minimum profit, review notes on May 8th to recall how to find the vertexMay 15th Notes: Solving Quadratic Problems Mixed Review ................
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