5.1 Solve System by graphing day 3 Date 1/14/19 Copy down Essential Question. Work on Warm Up. Essential Question Why can we call the solution to a system of equation the break even or the tie point? Warm Up: Find the slope and the y-intercept in each equation 1. 𝑦=− 3 5 𝑥+1 2. 𝑦=−𝑥 3. 𝑦− 1 2 𝑥=4 𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 =1 slope= − 3 5 𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 =0 slope= − 1 1 𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 =4 slope= 1 2

Practice: Solve system by graphing Review Problem Solve the system by graphing. Check your answer. 𝒚 = – 𝟓 𝟑 𝒙 –+𝟑 Graph the system. Find the slope and the y-intercept in each equation 𝒚 = 𝟏 𝟑 𝒙 −𝟑 The solution is (–2, 3).

Application Example 1 Wren and Jenni are reading the same book. Wren is on page 14 and reads 2 pages every night. Jenni is on page 6 and reads 3 pages every night. After how many nights will they have read the same number of pages? How many pages will that be?

Understand the Problem 1 Understand the Problem The answer will be the number of nights it takes for the number of pages read to be the same for both girls. List the important information: Wren on page 14 Reads 2 pages a night Jenni on page 6 Reads 3 pages a night

2 Make a Plan Write a system of equations, one equation to represent the number of pages read by each girl. Let x be the number of nights and y be the total pages read. Total pages is number read every night plus already read. Wren y = 2  x + 14 Jenni y = 3  x + 6

Solve 3 Graph y = 2x + 14 and y = 3x + 6. The lines appear to intersect at (8, 30). So, the number of pages read will be the same at 8 nights with a total of 30 pages.  (8, 30) Nights

  Look Back Check (8, 30) using both equations. 4 Check (8, 30) using both equations. Number of days for Wren to read 30 pages. 2(8) + 14 = 16 + 14 = 30  Number of days for Jenni to read 30 pages. 3(8) + 6 = 24 + 6 = 30 

Application Example 2 Video club A charges $10 for membership and $3 per movie rental. Video club B charges $15 for membership and $2 per movie rental. For how many movie rentals will the cost be the same at both video clubs? What is that cost?

Understand the Problem 1 Understand the Problem The answer will be the number of movies rented for which the cost will be the same at both clubs. List the important information: Rental price: Club A $3 Club B $2 Membership: Club A $10 Club B $15

2 Make a Plan Write a system of equations, one equation to represent the cost of Club A and one for Club B. Let x be the number of movies rented and y the total cost. Total cost is price for each rental plus member- ship fee. Club A y = 3  x + 10 Club B y = 2  x + 15

Solve 3 Graph y = 3x + 10 and y = 2x + 15. The lines appear to intersect at (5, 25). So, the cost will be the same for 5 rentals and the total cost will be $25.

  Look Back Check (5, 25) using both equations. 4 Check (5, 25) using both equations. Number of movie rentals for Club A to reach $25: 3(5) + 10 = 15 + 10 = 25  Number of movie rentals for Club B to reach $25: 2(5) + 15 = 10 + 15 = 25 

5.2 Solve System by Substitution Date 1/15/19 Copy down Essential Question. Work on Warm Up. Essential Question Why do you need two equations when trying to solve for two variables? Warm Up: Explain the word Word: Substitute Used it in a sentence and give one explain of how we use it in math.

Why can’t you solve 3𝑥+2𝑦=4 ? There is only 1 equation for 2 variable. You can you solve 2𝑦=4 There is 1 equation for 1 variable. For 3𝑥+2𝑦=4 ?How many equation do we need? There is 2 equation for 2 variable.

Rule of Linear Algebra There must be an equal amount of variable “unknowns” as equations for there to be a solution “ intersection” . Example. one equation, one unknown. two equations, two unknowns. three equations, three unknowns.

Example 1 : Solving a System of Linear Equations by Substitution Solve the system by substitution. y = 3x y = x – 2 Step 1 y = 3x Both equations are solved for y. y = x – 2 Step 2 y = x – 2 3x = x – 2 Substitute 3x for y in the second equation. Step 3 –x –x 2x = –2 2x = –2 2 2 x = –1 Solve for x. Subtract x from both sides and then divide by 2.

  Example 1 Continued Solve the system by substitution. Write one of the original equations. Step 4 y = 3x y = 3(–1) y = –3 Substitute –1 for x. Write the solution as an ordered pair. Step 5 (–1, –3) Check Substitute (–1, –3) into both equations in the system. y = 3x –3 3(–1) –3 –3  y = x – 2 –3 –1 – 2 –3 –3 

Example 1b : Solving a System of Linear Equations by Graphing Solve the system by graphing. y = 3x y = x – 2

5.2 Solve System by Substitution Date 1/15/19 Copy down Essential Question. Copy down and Work on Warm Up. Essential Question How does solving a system by substitute work? Explain the process in your own words. Warm Up: Find the cost of 1 apple and 1 orange

Practice 1: Solving a System of Linear Equations by Substitution Solve the system by substitution. y = x + 1 4x + y = 6 The first equation is solved for y. Step 1 y = x + 1 Step 2 4x + y = 6 4x + (x + 1) = 6 Substitute x + 1 for y in the second equation. 5x + 1 = 6 Simplify. Solve for x. Step 3 –1 –1 5x = 5 5 5 x = 1 5x = 5 Subtract 1 from both sides. Divide both sides by 5.

  Practice 1 Continued Solve the system by substitution. Write one of the original equations. Step 4 y = x + 1 y = 1 + 1 y = 2 Substitute 1 for x. Write the solution as an ordered pair. Step 5 (1, 2) Check Substitute (1, 2) into both equations in the system. y = x + 1 2 1 + 1 2 2  4x + y = 6 4(1) + 2 6 6 6 

Practice 2 : Solving a System of Linear Equations by Substitution Solve the system by substitution. x + 2y = –1 x = 5 + y Step 1 x + 2y = –1 Solve the first equation for x by subtracting 2y from both sides. (5+y) + 2y = –1 Step 2 x – y = 5 (–2y – 1) – y = 5 Substitute –2y – 1 for x in the second equation. –3y – 1 = 5 Simplify.

Practice 2 Continued Step 3 –3y – 1 = 5 Solve for y. +1 +1 –3y = 6 Add 1 to both sides. –3y = 6 –3 –3 y = –2 Divide both sides by –3. Step 4 x – y = 5 Write one of the original equations. x – (–2) = 5 x + 2 = 5 Substitute –2 for y. –2 –2 x = 3 Subtract 2 from both sides. Write the solution as an ordered pair. Step 5 (3, –2)

Practice 3 Solve the system by substitution. y = x + 3 y = 2x + 5 Step 1 y = x + 3 y = 2x + 5 Both equations are solved for y. Step 2 2x + 5 = x + 3 y = x + 3 Substitute 2x + 5 for y in the first equation. –x – 5 –x – 5 x = –2 Step 3 2x + 5 = x + 3 Solve for x. Subtract x and 5 from both sides.

Practice 3 Continued Solve the system by substitution. Write one of the original equations. Step 4 y = x + 3 y = –2 + 3 y = 1 Substitute –2 for x. Step 5 (–2, 1) Write the solution as an ordered pair.

Example 2 Solve the system by substitution. x = 2y – 4 x + 8y = 16 Step 1 x = 2y – 4 The first equation is solved for x. (2y – 4) + 8y = 16 x + 8y = 16 Step 2 Substitute 2y – 4 for x in the second equation. Step 3 10y – 4 = 16 Simplify. Then solve for y. +4 +4 10y = 20 Add 4 to both sides. 10y 20 10 10 = Divide both sides by 10. y = 2

Exmaple 2 Continued Solve the system by substitution. Step 4 x + 8y = 16 Write one of the original equations. x + 8(2) = 16 Substitute 2 for y. x + 16 = 16 Simplify. x = 0 – 16 –16 Subtract 16 from both sides. Write the solution as an ordered pair. Step 5 (0, 2)

Practice 4 Solve the system by substitution. 2x + y = –4 x + y = –7 Solve the second equation for x by subtracting y from each side. Step 1 x + y = –7 – y – y x = –y – 7 2(–y – 7) + y = –4 x = –y – 7 Step 2 Substitute –y – 7 for x in the first equation. 2(–y – 7) + y = –4 Distribute 2. –2y – 14 + y = –4

Practice 4 Continued Step 3 –2y – 14 + y = –4 Combine like terms. –y – 14 = –4 +14 +14 –y = 10 Add 14 to each side. y = –10 Step 4 x + y = –7 Write one of the original equations. x + (–10) = –7 Substitute –10 for y. x – 10 = – 7

Practice 4 Continued Step 5 x – 10 = –7 +10 +10 Add 10 to both sides. x = 3 Step 6 (3, –10) Write the solution as an ordered pair.

Tuesday 01/15/19 Homework Solution 1. x =5 y = 2x-3 Step 1 x=5 The first equation is solved for x. Step 2 y = 2(5)-3 y = 2x-3 Substitute x=5 for x in the second equation. Simplify. Step 3 y = 10-3 y = 7 Write the solution as an ordered pair. Step 4 (5, 7)

Tuesday 01/15/19 Homework Solution 2. x =y + 2 y = 2x-3 Step 1 x=y+2 The first equation is solved for x. Step 2 y = 2(y+2)-3 y = 2x-3 Substitute x=y +2 for x in the second equation. Simplify. Step 3 y = 2y + 4 -3 y = 2y +1 -2y -2y -1y = 1 y = -1

x =y + 2 Step 4 Substitute y = -1 for y back into the first equation. x =(-1) + 2 x = 1 Write the solution as an ordered pair. Step 5 (1, -1)

Tuesday 01/15/19 Homework Solution 3. 2x-3y=7 y=3x-7 Step 1 y=3x-7 The second equation is solved for x. Step 2 Substitute y = 3x-7 for u in the first equation. 2x - 3y =7 2x-3(3x-7) =7 2x-9x+21=7 Combine like terms Step 3 -7x+21=7 -21 -21 Simplify -7x=-14 −𝟕𝒙 −𝟕 = −𝟏𝟒 −𝟕 x=2

Step 4 y= 3x -7 Substitute x = 2 for y back into the second equation. y=3(2)-7 y=6-7 y=-1 Write the solution as an ordered pair. Step 5 (2, -1)

Wednesday 01/16/19 Homework Solution 1. 𝑥=−5 𝑦=2𝑥−3 Step 1 𝑥=−5 The first equation is solved for x. Step 2 𝑦=2𝑥−3 substitute 𝑥=−5 into the second equation 𝑦=2(−5)−3 Step 3 simplify to solve for y 𝑦=−10−3 𝑦=−13 Step 4 (−5, −13) write the solution as an order pair.

Wednesday 01/16/19 Homework Solution 2. 𝑥=−𝑦+2 𝑦=2𝑥−2 Step 1 𝑥=−𝑦+2 The first equation is solved for x. Step 2 𝑦=2𝑥−2 substitute 𝑥=−𝑦+2 into the second equation 𝑦=2(−𝑦+2)−2 Step 3 𝑦=−2𝑦+4−2 simplify to solve for y 𝑦=−2𝑦+2 +2𝑦 +2𝑦 3𝑦=2 3𝑦 3 = 2 3 𝑦= 2 3

Step 4 𝑥=−𝑦+2 substitute 𝑦= 2 3 into the first equation 𝑥=−( 2 3 )+2 𝑥= 4 3 Step 5 ( 4 3 , 2 3 ) write the solution as an order pair.

Wednesday 01/16/19 Homework Solution 3. 3𝑥+𝑦=9 𝑦=2𝑥+4 Step 1 𝑦=2𝑥+4 The second equation is solved for y. substitute 𝑦=2𝑥+4 into the first equation Step 2 3𝑥+𝑦=9 3𝑥+(2𝑥+4)=9 Step 3 3𝑥+2𝑥+4=9 simplify to solve for x 5𝑥+4=9 −4 −4 5𝑥=5 5𝑥 5 = 5 5 𝑥=1

Step 4 𝑦=2𝑥+4 substitute x=1 into the second equation and solved for y. 𝑦=2(1)+4 𝑦=2+4 𝑦=6 Step 5 (1, 6) write the solution as an order pair.

Thursday 01/17/19 Homework Solution 1. 𝑥−𝑦=2 𝑦−2𝑥=−3 Pre Step 1 𝑥−𝑦=2 do some algebra to solve for x +𝑦 +𝑦 𝑥=𝑦+2 Step 1 𝑦−2𝑥=−3 substitute x=y+2 into the second equation 𝑦−2(𝑦+2)=−3 Step 2 𝑦−2𝑦−4=−3 simplify to solve for y −1𝑦−4=−3 +4 +4 −1𝑦=1 𝑦=−1

Step 3 𝑥=𝑦+2 substitute y=−1 into the first equation and solved for x. 𝑥= −1 +2 𝑥=1 Step 4 (1, −1) write the solution as an order pair.

Thursday 01/17/19 Homework Solution 2. 2𝑥−3𝑦=7 𝑦−3𝑥=−7 Pre Step 1 𝑦−3𝑥=−7 do some algebra to solve for x +3𝑥 +3𝑥 𝑦=3𝑥−7 Step 1 2𝑥−3𝑦=7 substitute y=3x−7 into the first equation 2𝑥−3(3x−7)=7 Step 2 2𝑥−9𝑥+21=7 simplify to solve for x −7𝑥+21=7 −21 −21 −7𝑥=−14 −7𝑥 −7 = −14 −7 𝑥=2

Step 3 𝑦−3𝑥=−7 substitute x=2 into the first equation and solved for x. y−3(2)=−7 y−6=−7 +6 +6 𝑦=−1 𝑦=−1 Step 4 (2, −1) write the solution as an order pair.

Thursday 01/17/19 Homework Solution 3. 𝑥+𝑦=2 𝑦−2𝑥=−2 Pre Step 1 𝑥+𝑦=2 do some algebra to solve for y −𝑥 −𝑥 𝑦=−𝑥+2 Step 1 𝑦−2𝑥=−2 substitute y=−x+2 into the first equation (−x+2)−2𝑥=−2 Step 2 −𝑥+2−2𝑥=−2 simplify to solve for x 2−3𝑥=−2 −2 −2 −3𝑥=−4 −3𝑥 −3 = −4 −3 𝑥= 4 3

substitute 𝑥= 4 3 into the first equation and solved for x. Step 3 𝑥+𝑦=2 ( 4 3 )+𝑦=2 − 4 3 − 4 3 𝑦= 2 3 Step 5 ( 4 3 , 2 3 ) write the solution as an order pair.

.Make sure you submit the test when done. State Test Day Date 1/18/19 Drop off your homework to the back table Log into a laptop Cick on the CAASPP icon Log in with the info on the card I give you SESSION ID: .Make sure you submit the test when done. I will give out a card with your log in info *no notes *you can use scratch paper *you can have all the time you need, but only TODAY * this goes in the grade as a test score