Algebra 4

Jonathan uploaded some original songs and also some pictures. Alissa uploaded four times as many songs, but only uploaded twice as many pictures. Jonathan uploaded 11 items while Alissa uploaded 24. How many songs did Jonathan upload?

1

What exponential function represents the data in the table?

f(x) = 4x

Brad can make 4 key chains in an hour. Velma can make only 3 key chains in an hour, but she already has 6 completed key chains. Explain to Brad how he can use a system of equations to determine when he will have the same number of key chains as Velma. Use complete sentences.

let t= time in hours, b= no of keychains made by brad, v=no of keychains made by velma. b=4t v=3t+6 equate b and v to get time t when they have equal keychains. so, 4t=3t+6 so, t=6 hours

At the end of April, Mandy told Bill that she has read 16 books this year and reads 2 books each month. Bill wants to catch up to Mandy. He tracks his book reading with a table on his door. Using his table below, what month will Bill have read the same amount of books as Mandy?

December

Samantha is going on vacation for the summer and is trying to choose between two different plans. The first plan costs $450 for 3 days at a hotel and 2 days at an amusement park. The second plan offers 5 days at the same hotel and 3 days at the amusement park for $700. The cost of 1 day at the hotel and the amusement park is the same under both plans. How much does a 1-day trip to the amusement park cost?

$150 WORK ON PAPER

If $360 is invested at an interest rate of 4% per year and is compounded quarterly, how much will the investment be worth in 18 years?

$736.96

Solve the following system of equations: 2x + 3y − z = 1 3x + y + 2z = 12 x + 2y − 3z = −5

(3,-1,2)

Solve the following system of equations by graphing and select the correct answer below: 3x + 5y = 38 4x − 2y = 16

(6,4)

Solve the following system of equations by substitution and select the correct answer below: 6x − 4y = 36 2x − 8y = 32

6x - 4y = 36 2x - 8y = 32 ( multiply by '-3'( . Hence 6x - 4y = 36 -6x + 24y = -96 Add 20y = -60 y = -3 Hence 6x - 4(-3) = 36 6x + 12 = 36 6x = 24 x = 4 Hence Answer is (4, -3).

What is one of the solutions to the following system of equations? y2 + x2 = 53 y − x = 5

Answer: B. (−7, −2) SOLVINGS y^2 + x^2 = 53 ..... (Equation I) y − x = 5 ..... (Equation II) Make y the subject of equation II y = x + 5 ..... (Equation III) Substitute (x + 5) for the value of y in Equation I ∴ (x + 5)^2 + x^2 = 53 x^2 + 10x + 25 + x^2 = 53 2 x^2 + 10x + 25 = 53 2 x^2 + 10x + 25 - 53 = 0 2 x^2 + 10x - 28 = 0 ..... (Equation IV) Factor 2 out of equation IV; 2 (x^2 + 5x - 14) = 0 ..... (Equation V) Divide both sides of equation V by 0 ∴ x^2 + 5x - 14 = 0 x^2 + (7x - 2x) - 14 = 0 x^2 + 7x - 2x - 14 = 0 ..... (Equation VI) Divide Equation VI into two groups (x^2 + 7x and - 2x - 14) x + 7 is a common factor in both groups Factor out x + 7 in Equation VI ∴ x(x + 7) - 2(x +7) = 0 (x - 2) (x +7) = 0 ∴ x - 2 = 0 OR x + 7 = 0 x = 2 OR x = -7 x = 2, -7 If x = 2 Substitute 2 for the value of x in equation II y − x = 5 y - 2 = 5 y = 5 +2 y = 7(x =2, y =7) ... Not included in the options If x = -7 Substitute -7 for the value of x in equation II y − x = 5 y - (-7) = 5 y + 7 = 5 y = 5 - 7 y = -2(x = -7, y = -2) .... This tally with option B

Choose the graph that matches the following system of equations: 8x + 5y = −5 7x − 3y = 3

In intercept form, the equations are .. x/(-5/8) +y/(-1) = 1 .. x/(3/7) +y/(-1) = 1 The first line has intercepts (-5/8, 0) and (0, -1). The second line has intercepts (3/7, 0) and (0, -1) In the second graph, the (blue) line with positive slope has an x-intercept greater than 1/2. Neither line matches that. The intercepts match those of the first graph.

Your boss hands you the monthly data that show the number of orders coming in to and out of the warehouse. The data are in the table below. Explain to your boss, in complete sentences, the solution to this system and what the solution represents.

Month No. of orders in No. of orders out January (1) 3 1 February (2) 4 3 March (3) 5 5 April (4) 6 7 Number of orders in a = number of month + 2 f(x) = x + 2. Number of orders out = 2 number of month + 1 g(x) = 2x - 1 So, in words, the numbers of orders is equal to the number of month plus 1. This represents a linear function, with slope 1 and y-intercept 2. And the numbers of orders out is also a linear function with slope 2 and y-intercept - 1.

Solve 1 / 36 = 6x−3

x=1

How can one half x − 5 = one third x + 6 be set up as a system of equations?

y = (1/2)(x) - 5 multiply each term by 2 to clear the fraction 2y = x - 10 y = (1/3)(x) + 6 multiply each term by 3 to clear the fraction 3y = x + 18 2y = x - 10 Rearranging 2y - x = -10 3y = x + 18 3y - x = 18

Which system of equations is represented by the graph? A Rational graph with asymptotes of x equals negative 2 and y equals 1. Linear graph intersects it at (-1, -5) and (4, 0).

y = x - 4/ x + 2 y = x − 4

Which system of equations is represented by the graph? (4,0) (-1,-5)

y = x - 4/ x + 2 y = x − 4

Which system of equations is represented by the graph? (6, 9) (-5, 3).

y = x² + 5x + 3 6x + y = −27

What is the solution to the following system of equations? y = x2 + 10x + 11 y = x2 + x − 7

y = y x^2+10x+ 11 = x^2+x -7 10x-x = -7-11 9x=-18 x=-2 y = (-2)^2+(-2)-7 = 4-2-7 =-5 (-2,-5)

What is the solution to the following system of equations? y = x2 + 10x + 11 y = x2 + x − 7

y=x^2+10x+11 y=x^2+x-7 therefore, x^2+10x+11=x^2+x-7 x^+10x+11-x^2-x+7=0 9x+18=0 x=-2 y=(-2)^2+10(-2)+11= -5

What are the solutions to the following system of equations? y = x2 + 3x − 7 3x − y = −2

y = 3x + 2 3x + 2 = x^2 + 3x - 7 x^2 -9 = 0 x^2 = 9 x1 = 3 x2 = -3 y1 = 3*3+2 = 11 y2 = -3*3 + 2 = -7 (3,11) and (-3,-7)

What are the solutions to the following system of equations? y = x2 + 3x − 7 3x − y = −2

y = 3x + 2 after that we get: 3x + 2 = x^2 + 3x - 7 x^2 -9 = 0 x^2 = 9 x1 = 3 x2 = -3 y1 = 3*3+2 = 11 y2 = -3*3 + 2 = -7 we get pairs: (3,11) and (-3,-7)

Solve 81x = 27x + 2

x=6

A jet ski depreciates at 11% of its original value each year. If the jet ski was $8,000 at its time of purchase, what is the value of the jet ski after 5 years?

$4,467.25

If $535 is invested at an interest rate of 6% per year and is compounded continuously, how much will the investment be worth in 10 years?

$974.83

Solve the following system using elimination: 3x + 5y = 1 4x + 2y = -8

(-3, 2)

Solve the following system using elimination: 4x - y = 5 x + 5y = -4

(1, -1)

What is the solution to the following system of equations? y = −x² − 5x − 4 y = −x² + 9x − 18

(1, −10)

In the below system, solve for y in the first equation. x − 3y = −6 2x − 7y = 10

(1/3)x + 2 WORK ON PAPER

Which system of equations is represented by the graph? a line is graphed through points negative (6, 9) and (-5, 3). It intersects a parabola that opens up at these two points. Courtesy of Texas Instruments

y = x² + 5x + 3 6x + y = −27

If you were to use the substitution method to solve the following system, choose the new equation after the expression equivalent to y from the first equation is substituted into the second equation. 6x − y = 1 4x − 3y = −11

6x - y = 1 6x - y - 1 = 0 6x - 1 = y Substituting we get 4x - 3(6x - 1) = -11

Solve 3x − 5y = 17 9x − 15y = −4

No solutions

Solve the following system: x²+y² = 17 y = x - 3

(4, 1) , (-1, -4)

Solve the following system using substitution: x - 4y = 1 3x - y = 14

(5, 1)

Solve 7x − 2y = −3 14x + y = 14

(5/7, 4)

Solve the following system using elimination: 2x - 2y = 10 3x + 2y = 20

(6, 1)

What is one of the solutions to the following system of equations? y² + x² = 65 y + x = 7

(8, −1)

Solve 5x − 6y = −38 3x + 4y = 0

(−4, 3)

What is the value of y in the solution to the following system of equations? 5x − 3y = −3 2x − 6y = −6

1

A bacteria culture begins with 4 bacteria which double in size every hour. How many bacteria exist in the culture after 8 hours?

1,024

If a hospital patient is given 50 milligrams of medicine which leaves the bloodstream at 10% per hour, how many milligrams of medicine will remain in the system after 6 hours? Use the function A(t) = Ie^rt.

27.44 mg

Solve the following system of equations by graphing and select the correct answer below: 2x + 6y = 20 3x − 2y = 8

2x+6y+(9x-6y)=20+24 11x=44 X=4 Y=2

What are the solutions to the following system of equations? Select the correct answer below. y = x2 − 2x + 4 8x + y = 20

(2, 4) and (−8, 84)

Solve the following system: y = x² + 2x - 3 -3x + y = -1

(2, 5), (-1, 4)

Solve the following system using substitution: 2x - 4y = 6 3x + 9y = 9

(3, 0)

How can 1/3x − 2 = 1/4x + 11 be set up as a system of equations?

1/3x - 2 = 1/4x + 11....it looks to me like one y was subbed in for another... ur 2 equations are : y = 1/3x - 2 and y = 1/4x + 11...but we need them in standard form... y = 1/3x - 2 -1/3x + y = -2 ..multiply by 3 -x + 3y = -6.....or 3y - x = -6 <== y = 1/4x + 11 -1/4x + y = 11 ...multiply by 4 -x + 4y = 44 or 4y - x = 44

Gina, Sam, and Robby all rented movies from the same video store. They each rented some dramas, comedies, and documentaries. Gina rented 11 movies total. Sam rented twice as many dramas, three times as many comedies, and twice as many documentaries as Gina. He rented 27 movies total. If Robby rented 19 movies total with the same number of dramas, twice as many comedies, and twice as many documentaries as Gina, how many movies of each type did Gina rent?

3 dramas, 5 comedies, and 3 documentaries WORK ON PAPER

What equation is solved by the graphed systems of equations? (7,25)

3x + 4 = 5x − 10

Solve the following system of equations by substitution and select the correct answer below: 4x − 6y = 28 8x − 4y = 24

4x - 6y = 28......(1) 8x - 4y = 24......(2) Using Equation (2) 8x - 4y = 24 8x = 24 + 4y x = 3 + 1/2y....(2) Substituting x = 3 + 1/2y into Equation (1) 4(3 + 1/2y) - 6y = 28 12 + 2y - 6y = 28 Collect like terms: 2y - 6y = 28 - 12 -4y = 16 y = -4 Substitute y = -4 into Equation (1) 4x - 6y = 28 4x - 6(-4) = 28 4x + 24 = 28 4x = 28 - 24 4x = 4 x = 1 x = 1; y = -4

Use the substitution method to solve the following system of equations: 4x − y = 3 7x − 9y = −2

4x = y + 3 ---> y = 4x - 3 7x - 9(4x-3) = -2 ---> 7x - 36x + 27 = -2 -29x = -29 x = 1 (1, 1)

How can 1/4x − 3 = 1/2x + 8 be set up as a system of equations?

4y − x = −12 2y − x = 16

What is the value of y in the solution to the following system of equations? 5x − 3y = −11 2x − 6y = −14

5x − 3y = −11 x2 2x − 6y = −14 x2 (-5) 10x - 6y = -22 -10x + 30y - 70 ------------------ 30y - 6y = 70 - 22 24y = 48 y = 2

What type of solutions do the methods for solving systems of equations find and how does this relate to setting the equations equal to each other?

A system of equations is a set of two or more equations that share two or more unknowns. The solutions to a system of equations are the total amounts that make all of the equations true, as well as the locations where the graphs of the equations intersect. You can determine a system of linear equations through graphing, linear combination and substitution. Systems of nonlinear functions, for example, quadratic or exponential equations, can be managed with the equivalent methods.

What are the solutions to the following system of equations? y = x² + 12x + 30 8x − y = 10

No real solutions

The Mathalot Company makes and sells textbooks. They have one linear function that represents the cost of producing textbooks and another linear function that models how much income they get from those textbooks. Describe the key features that would determine if these linear functions ever intercepted.

If two lines have the same slope, they are parallel and will never intercept. If they don't have the same slope, they will intercept.

Solve the system y = 2x + 6 and 3y = 6x + 18 by using graph paper or graphing technology. What is the solution to the system?

Infinite Solutions

Solve 5x + 2y = 7 10x + 4y = 14

Infinitely many solutions

Mark went to an electronic store, purchased 2 CDs and 6 DVDs, and spent $156. Antonio went to the same store, bought 1 CD and 9 DVDs, and spent $210. How much does each CD sell for at the store?

Let C be CD's Let D be DVD's 2C + 6D = 156 C + 9D = 210 6D = 156 - 2C 3D = 78 - C C + 9D = 210 C + 3(78 - C) = 210 C + 234 - 3C = 210 234 - 210 = 3C - C 24 = 2C 12 = C The cost of one CD is $12

Amy purchased 17 pencils and 18 pens for a fund-raiser at school and spent $61.50. Jocelyn purchased 10 pencils and 21 pens and spent $57. How much does each pencil cost?

Let, pencils be "x" pens be "y" According to the question, 17x + 18y = 61.50 ----------> equation (1) 10x + 21y = 57 -----------------> equation (2) Now, Taking equation (1), 17x + 18y = 61.50 18y = 61.50 - 17x y = (61.50 - 17x ) / 18 -----> equation (3) Now, substituting the value of "y" in equation (2), we get, 10x+21 (61.50-17x/18)=57 10x+7 (61.50-17x/6)=57 10x+ 430.50-119x/6=57 10x*6/6 430.50-119x/6=57 60x/6+ 430.50-119x/6=57 430.50+60x-119x/6=57 430.50-59x=342 59x=430.50-342 x=88.5/59 x=1.5 So, Each pencil costs $1.5

What is the graph of the system y = −2x + 3 and 2x + 4y = 8?

Line through point (0,3) and (-1,1). Line through (0,2) and (4,0) The one that resembles this the most is letter C

Mark, Jessica, and Nate each downloaded music from the same website. Mark downloaded 10 songs in total consisting of pop, rock, and hip hop. Jessica downloaded five times as many pop songs, twice as many rock, and three times as many hip hop songs as Mark. She downloaded 28 songs total. If Nate downloaded 20 songs total with three times as many pop songs, three times as many rock songs, and the same number of hip hop songs as Mark, how many songs of each type did Mark download?

Mark downloaded 1 pop songs, 4 rock songs, and 5 hip hop songs.

Solve the following system of equations and show all work. y = 2x2 y = −3x −1

Since both right sides of each equation is equal to y, we can set them equal to each other. 2x² = -3x - 1 Let's get everything on the left side... 2x² + 3x + 1 = 0 Let's solve this quadratic by splitting the middle... We want two #'s that add up to 3 and multiply to 2. (2×1) These are 2 and 1. Let's split the middle into 2x and x. 2x² + 2x + x + 1 = 0 Factor the first and last two terms. 2x(x+1)+1(x+1) = 0 (2x+1)(x+1) = 0 If any value of x makes either factor evaluate to 0, it is a solution. 2x + 1 = 0 ⇒ 2x = -1 ⇒ x = -1/2 ⇒ y = 2(-1/2)² = 2(1/4) = 1/2 x + 1 = 0 ⇒ x = -1 ⇒ y = 2(-1)² = 2(1) = 2 Our solutions are (-1/2, 1/2) and (-1, 2).

What are the solutions to the following system of equations? Select the correct answer below. y = x2 − 4x + 3 2x + y = 6

Substitute x2−4x+3 for y into 2x+y=6 then solve for x. x=3,−1Substitute 0 for x into 2x+y=6 then solve for y. y=0Substitute 8 for x into 2x+y=6 then solve for y. y=8 The solution can be shown in both point and equation forms. Point Form: (3,0),(−1,8) Equation Form: x=3,−1 y=0,8

How do you use properties of exponents and logarithms to rewrite functions in equivalent forms and solve equations?

The exponential function of the base a, so the generic form is f (x) = a ^ x, as a positive number beside one. Every exponential function of form f (x) = a^x, act with these Properties - 1. The function applied to the zero value is always equal to 1: f (0) = a ^ 0 = 1 2. The exponential function of 1 is always equal to the base: f (1) = a ^ 1 = a. 3. The exponential function of a total is equal to the product of the use of the function on each value separately. f (m + n) = a ^ (m + n) = a ^ m · a ^ n = f (m) · f (n). 4. The exponential function of subtraction is equal to the quotient use to a quantity or number which another is to be subtracted, then divided by the application to the quantity or number to be subtracted from another: f (p - q) = a ^ (p - q) = a ^ p / a ^ q Logarithm: In the log (b), a is called the base of the logarithm and b is called an argument, with a and b positive. Consequently, the definition of logarithm is: log b = n --> a ^ n = b (a > 0, b > 0)

Which equation does the graph of the systems of equations solve?

The solution to this system is (x,y) = (2,-3) D. one half x − 4 = −2x + 1 >>> x/2 - 4 = y...........> (2)/2 -4 = -3 >>> -2x + 1 = y ......> -2(2) +1 = -3

Determine whether the point (1, 1) is a solution to the system of equations. Explain your reasoning in complete sentences.

The solutions to the system of equations are the point of intersection of the graphs of the equations. The point of intersection of the graphs of the given equation is (0, 2). Therefore, point (1, 1) is not a solution to the system of equation.

Solve the following system of equations: 4x + 7y − 2z = 0 3x − 5y + 3z = 9 3x + 6y − z = 1

[1] 4x + 7y - 2z = 0 [2] 3x - 5y + 3z = 9 [3] 3x + 6y - z = 1 Solve by Substitution : // Solve equation [3] for the variable z [3] z = 3x + 6y - 1 // Plug this in for variable z in equation [1] [1] 4x + 7y - 2•(3x+6y-1) = 0 [1] -2x - 5y = -2 // Plug this in for variable z in equation [2] [2] 3x - 5y + 3•(3x+6y-1) = 9 [2] 12x + 13y = 12 // Solve equation [2] for the variable y [2] 13y = -12x + 12 [2] y = -12x/13 + 12/13 // Plug this in for variable y in equation [1] [1] -2x - 5•(-12x/13+12/13) = -2 [1] 34x/13 = 34/13 [1] 34x = 34 // Solve equation [1] for the variable x [1] 34x = 34 [1] x = 1 // By now we know this much : x = 1 y = -12x/13+12/13 z = 3x+6y-1 // Use the x value to solve for y y = -(12/13)(1)+12/13 = 0 // Use the x and y values to solve for z z = 3(1)+6(0)-1 = 2 Solution : {x,y,z} = {1,0,2}

Marcia can make 5 candles in an hour. Kevin can make only 4 candles in an hour, but he already has 7 completed candles. Explain to Marcia how she can use a system of equations to determine when she will have the same number of candles as Kevin. Use complete sentences.

m(h)=5h that is Marcia's number of candles where she makes 5 candles per hour (h) kevin makes 4 but already did 7 so k(h)=4h+7 when will they have the same number of candles? when will m(h)=k(h) set equal 5h=4h+7 minus 4h from both sides 1h=7 answer is 7 hours the system of equation is m=5h k=4h+7 find what value of h makes m=k true answer is 7 hours

Solve the system 2x + 3y = 3 and 3x − 2y = 11 by using graph paper or graphing technology. What is the solution to the system?

two linear equations 2x + 3y = 3 and 3x - 2y = 11 and is asked for the solution of the system of equations. In this case, we can eliminate x by multiplying the second equation by -2/3. This leaves y to be equal to -1 and x from the first eqn to be equal to 3. (3, -1)

If you were to use the substitution method to solve the following system, choose the new equation after the expression equivalent to x from the first equation is substituted into the second equation. x + 4y = −9 2x + 5y = −6

x + 4y = -9 2x + 5y = -6 x = -4y - 9 plug into the 2x + 5y = -6 in place of the 'x'. 2(-4y -9) + 5y = -6

Sherri has 23 pieces of jewelry to sell. She sells the bracelets for $2 and the necklaces for $3, and earns a total of $55. If the bracelets are represented by x and the necklaces are represented by y, which of the following systems of equations can be used to calculate the number of bracelets and necklaces sold?

x = bracelets y = necklaces there are 23 of them in total (necklaces + bracelets) x + y = 23 bracelets are $2 each and necklaces are $3 each...totaling $55 2x + 3y = 55

Kara has 21 collectible buttons to sell. She sells the small buttons for $3 and the large buttons for $4, and earns a total of $68. If the small buttons are represented by x and the large buttons are represented by y which of the following systems of equations can be used to calculate the number of large and small buttons sold.

x+y=21 3x+4y=68

Solve 8 = 2x + 4.

x=-1

Solve (square root of 7) ^6 x = 49^x−6

x=-12