euph0ria

let's set up two equations, and define two variables. we'll go with x and y again.

here is the system of equations:

x + y = 11

2x - y = 7

again (assuming you are the same Guest who asked a similar question a couple minutes before), i will use elimination.

2x + 2y = 22

2x - y = 7

now, let's subtract these two equations.

3y = 15

y = 5

and again, let's solve for x by plugging in the y value in one of the initial equations.

x + 5 = 11

x = 11 - 5

x = 6

hope this helped! please let me know if you are confused about anything i did :)

let's define two variables for the first and second numbers: x and y. 

now we can create a system of equations -

x - y = 4.

2x + 3y = 28.

i'm going to use elimination for this one, though substitution works just as well! i will multiply the first equation by 2.

2x - 2y = 8

2x + 3y = 28

so now we can subtract these two equations.

5y = 20

y = 4

to solve for x, we plug in the value for y in one of the initial equations.

x - 4 = 4

x = 8

and we're done! hope this helped! please let me know if you are confused about anything i did 

because the lines are parallel, we can conclude that angle 3 + angle 5 is 180 (they both make up a straight angle).

now we can make this equation:

5x + 10 + 3x + 10 = 180

8x + 20 = 180

8x = 160

x = 20

hope this helped! sorry if my explanations weren't too clear. feel free to ask any questions you have about what i did! :)

let's define a variable p for pencils and e for erasers.

now we can set up a sysem of equations.

2p + 3e = 140

and

5p + 4e = 280

for this problem, i'm going to use elimination.

we multiplfy the first equation by 5 and the second equation by 2.

10p + 15e = 700

10p + 8e = 560

notice how we now have a 10p on both equations. we can subtract the two equations to cancel those out.

7e = 700 - 560

7e = 140

e = 20

now we can solve for p.

2p + 3(20) = 140

2p + 60 = 140

2p = 80

p = 40

hope this helped! please let me know if you are confused about anything i did 

the slope-intercept form for any line is y = mx + b, where m is the slope and b is the y-intercept.

now, let's calculate the slope:

\(\frac{1-2}{7-8}\)

= \(\frac{-1}{-1} = 1\)

here is the equation we currently have solved: y = x + b

now we have to solve for the y-intercept. to do this, we substitute one of the given points into the equation, and solve for b.

let's use (8, 2). in this ordered pair, the 8 is the x, and the 2 is the y.

2 = 8 + b

2 - 8 = b

b = -6

and now we have our final equation! 

y = x - 6

hope this helped! please let me know if you are confused about anything i did 

here are the two equations you can write for this one:

x = 50 + y ("There were 50 more adult tickets sold compared to student tickets.")

5x + 2y = 950 ("Adult tickets were 5 dollars, student tickets were 2 dollars, and they collected a total of 950 dollars.")

now, we can use substitution and simplify.

5(50 + y) + 2y = 950

250 + 5y + 2y = 950

7y + 250 = 950

7y = 700

y = 100

now we should plug this variable back in an equation, to solve for x.

x = 50 + 100

x = 150

hope this helped! please let me know if you are confused about anything i did :)

i believe you are the same Guest as the one who asked a very similar question, except about parallelograms. 

so, again, i'm not sure how to explain this, but i believe the answer is option 3.

here is a site that has more information on rhombuses: 

https://byjus.com/maths/rhombus/

hope this helped! if you are interested in why 1, 2, or 4 are all properties of rhombuses, feel free to ask here, check the site i linked above, or do a quick internet search! 

the volume of the larger prism is 8*9*5 = 360.

the volume of the small cube is 2^3 = 8.

the area that is in the large prism but not in the cube is just the volume of the larger prism minus the volume of the cube.

so, the answer would be 360 - 8 = 352. 

i see that your initial answer was 64? if you like, feel free to explain how you got that as your answer, and i can help you see what you did wrong 

hope this helped!

a) yes, the two rectangles are similar. this is because PQRS is made by applying a scale factor to ABCD, meaning that each side of PQRS has a constant ratio to ABCD. 

(yikes, sorry if this explanation wasn't too good)

b) the area is simply length * width, which is 9*4 = 36.

c) the side lengths of PQRS are 9*5 = 45, and 4*5 = 20. the area of PQRS is 45*20 = 900.

d) the ratio of the area of ABCD to the area of PQRS is 36/900 = 1/25. this means that the area of PQRS is 25 times that of ABCD, which is also the square of the scale factor.

hope this helped! again, sorry if my explanations weren't that good. feel free to ask any further questions :)

we can set up equations for this one. let's define a variable x, to be the first piece of pipe cut.

so, we know the sum of all the pieces is 84.

x {this is for the first pipe} + x - 3 {this is for the second pipe} + x + 4 {this is for the third pipe} = 84.

now, we can combine the like terms, and then isolate the variables.

3x + 1 = 84

3x = 84 - 1

3x = 83

x = 83/3

well, the numbers did not end up quite as nice, but we can still solve for each piece.

the first piece is just 83/3.

the second piece is 83/3 - 9/3 = 74/3.

the third piece is 83/3 + 12/3 = 95/3.

if you would like to double check, you can add up all of those and see if they add up to 84!

hope this helped! please let me know if you are still confused