Algebra 4

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

Infinitely many solutions

$150 WORK ON PAPER

(1/3)x + 2 WORK ON PAPER

No solutions

1

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

(−4, 3)

(5/7, 4)

1

(6, 1)

(1, -1)

(-3, 2)

(5, 1)

(3, 0)

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

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

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

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

(8, −1)

y = x² + 5x + 3 6x + y = −27

(1, −10)

No real solutions

y = x - 4/ x + 2 y = x − 4

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)

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)

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)

Infinite Solutions

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

December

3x + 4 = 5x − 10

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

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

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

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

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

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

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

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

[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}

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.

4y − x = −12 2y − x = 16

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

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)

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

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

y = x - 4/ x + 2 y = x − 4

y = x² + 5x + 3 6x + y = −27

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).

(2, 4) and (−8, 84)

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

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

(6,4)

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

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

(3,-1,2)

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

f(x) = 4x

x=-1

x=6

x=1

x=-12

$736.96

$974.83

1,024

$4,467.25

27.44 mg

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.

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.

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).

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.

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.

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)