125- week 12- Algebra I pg.169-
-2x - y = -3 3x - y = 17
-2x - y = -3 3x - y = 17 (-1) -2x - y = -3 ------------------- 2x +y = 3 3x - y = 17 5x =20 x = 4 -2(4) - y = -3 -8 -y = -3 -y = 5 y = -5 (4,-5) 3(4) - -5 = 17 12 +5 = 17 17 = 17 -2(4) - -5 = -3 -8 +5 = -3 -3 = -3
A number is one more than twice as large as another number. Twice the smaller number added to three times the larger number is 59 X= the smaller number Y= The larger number Y=2x+1
-2x+y=1 2x+3y=59 —————— 4y=60 /4. /4 Y=15 -2x+15=1 -2x=-14 /-2. /-2 X=7 7 and 15 -2(7)+15=1 -14+15=1 1=1 2x+3y=59 2(7)+3(15) =59 14+ 45=59 59=59
Add vertically (-3x²-6x+8)+(8x²+7x-14)
-3x² -6x +8 8x² +7x -14 ------------------ 5x² +x - 6
2x - y = -9 x + 2y =-7
2x - y = -9 x + 2y = -7 ------------- (2) 2x - y = -9 x + 2y = -7 4x - 2y = -18 x + 2y = -7 --------------- 5x = -25 5x / 5 = -25/5 x = -5 2(-5) + -y = -9 -10 + -y = -9 -10 + 9 = y -1 = y check it out 2(-5) + --1 = -9 -10 - -1 = -9 -10 + 1 = -9 -9 = -9 -5 + 2(-1) = -7 -5 + - 2 = -7 -7 = -7 -----------------
Solve by elimination- 3x+4y=6 -3x+2y=8
3x +4y=6 -3x+2y=8 ——————— 6y=14 6y/6=14/6 or 7/3 Y=7/3 3x +4(7/3) =6 3x +28/3=6 3x =6/1 + -28/3 3x=18/3 + -28/3 3x = -10/3 3x/3 = -10/3 times 1/3 X= -10/9 (-10/9, 7/3) Check 3(-10/9) +4(7/3)=6 -30/9 + 28/3=6 -30/9 + 84/9= 6 54/9=6 6=6 -3(-10/9) + 2(7/3)=8 30/9 + 14/3=8 30/9 + 42/9= 8 72/9=8 8=8
solve by elimination 3y = 18-x 7x = 11 + 2y
3y = 18-x 7x = 11 + 2y -------------- rearrange 3y + x = 18 -2y + -7x = 11 ----------------- (2) 3y + x = 18 (3) -2y + -7x = 11 6y + 2x = 36 -6y +-21x = 33 23x = 69 23x/23 = 69/23 x = 3 3y = 18 -3 3y = 15 3y/3 = 15/3 y = 5 check it out 3(5) = 18 -3 15 = 15 7(3) = 11 + 2(5) 21 = 11 + 10 21 = 21
3x + 2y = 6 -3x + y =3
3y = 9 /3 /3 y = 3 3x + 2(3) = 6 3x + 6 = 6 x = 0 -3(0) + y = 3 0 + y=3 y = 3 (0,3) 3(0) + 2 ( 3 ) = 6 6 = 6
solve by elimination 5x + 3y =16 3x -5y = -4
5x + 3y = 16 3x - 5y = -4 (5) 5x +3y=16 (3) 3x -5y=-4 25x +15y= 80 9x -15y=-12 34x =68 34x/34 = 68/34 x =2 solve for y 5(2) +3y = 16 10 + 3y = 16 3y = 6 y =2 x=2 y=2 check 5(2) +3(2) = 16 10 + 6 = 1 16=16 3(2) + -5(2) = -4 6 + -10 = -4 -4=-4
add - 5x -10y +3y -7z +5x-10y-3z-11x+4z+7x-9z
5x+5x-11x+7x -10y+3y-10y-7z-3z+4z-9z =6x -17y -15z
add- 5(x²+x-5) +2(3x² -2x -7)
5x² +5x -25 +6x²-4x -14 11x² +x+-39
Add vertically (5x³+8x + 2) + (3x² - 2x +1)
5x³ + 0x² + 8x + 2 0x³ +3x² -2x +1 ------------------- 5x³ + 3x² +6x + 3
Add horizontally (5x³ +8x²-3x+1)+(x³-4x-5)
5x³ +x³ +8x² +-3x +-4x +1 -5 6x³ + 8x² -7x -4
Add horizontally (1 - 3q²+8q³) +3 - q²+1-4q³
8q³ -4q³ +3q² +q² +1 + 3 + 1 4q³ +4q²+5
The sum of a two digit number is 13. When the digits are reversed, the new number is 27 more than the original number. What is the original number?
X +y=13 10x + y= 10y +x +27 —————————— 9x = 9y +27 X + y =13 9x +-9y=27 X+y=13 (9) ————- 9x +-9y=27 9x + 9y = 117 18x=144 X=8 Y= 5 Original number is 58 58+27=85
One number is 5 more than twice as much as another number. Thes sum of the numbers is 56. Find the numbers X= smaller number Y= larger number X+y=56 Y=2x+5
X+y=56 -2x+y=5 X+y=56 2x+-y=-5 3x=51 3x/3=51/3 X=17 Y+17=56 Y=39 17 and 39 39=2(17)+5 39= 34+5 39=39
Use substitution Find two numbers whose difference is 38 And whose sum is 70
X+y=70 X-y=38 ------------ Y=70-x X-(70-x)=38 X-70+x=38 2x=108 2x/2=108/2 X=54 54+y=70 Y=70-54 Y=16 The two numbers are 16 and 54 Check 54+16=70 70=70 54-16=38 38=38
The sum of two numbers is 71 and their difference is 13. Find the numbers
X+y=71 X-y=13 ------------- 2x = 84 2x/2=84/2 X=42 42+y=71 Y=71-42 Y=29 The two numbers are 42 and 29 Check42+29=71 71=71 42-29=13 13=13
The sum of two numbers is 48 and their difference is 14. What are the two numbers?
X=larger Y=smaller X+y=48 X-y=14 ----------- 2x =62 2x/2=62/2 X=31 31+y=48 Y=48-31 Y=17 The two numbers are 31 and 17 Check 31+17=48 48=48 31-17=14 14=14
Cashews prices at $10.00 per pound are to be mixed with almonds prices at $7.00 per pound to make 75 pounds of mixed nuts to sell at $8.20 per pound. How many pounds of cashews and how many pounds of almonds should be used?
a+c=75 (10)c + (7)a= (8.20)75 10c + 7a = 615 --------------- (-7) c + a= 75 10c + 7a = 615 -7c + -7a = -525 ------------------------ 3c = 90 /3 /3 c = 30 10(30) +7a = 615 300 +7a = 615 7a = 615 - 300 7a = 315 /7 /7 a =45 30 + 45 = 75 30 pounds of cashew 45 pounds of almonds
The sum of the digits of a two-digit number is 10. If the number is written in reverse, it is 54 more than the original number. Find the number!
x + y = 10 10x + y = 10y +x + 54 9x + -9y = 54 x + y = 10 (9) 9x + 9y =90 9x - 9y =54 18x =144 /18 /18 x = 8 y = 2 82 = 28 +54 82 = 82
The sum of the digits of a two-digit number is 13. If the number is written in reverse, it is 9 more than the original number. Find the number.
x + y = 13 10x +y = 10y + x +9 9x +-9y =9 x + y =13 x + y = 13(9) 9x +9y =117 9x - 9y =9 18x = 126 /18 /18 x =6 y = 7 67 + 9 =76 76 = 76 67 and 76
How many pounds each of dried fruit selling for $3.00 per pound and $3.80 per pound must be mixed to make a mixture of 80 pounds to sell at $3.50 pound x = priced at $3.00 y = priced at $3.80
x + y = 80 3(x) + 3.8(y) = 3.50(80) ------------------ 3x + 3.8y = 280 ( -3)x + y = 80 ---------------------- -3x +-3y = -240 3x + 3.8y = 280 .8y =40 /8 /8 y = 50 x + 50 = 80 x = 30 check it - 3(30) + 3.8(50) = 280 90 + 190 = 280
The size of two complementary angles is in the ratio of 1:3 find the two angles (complementary angles have a sum of 90 degrees)
x + y = 90 3x = y x +y = 90 3x +-y = 0 ---------------------- 4x =90 /4 /4 x =22.5 22.5 + y = 90 y = 67.5 3( 22.5) =67.5
