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Find all real or imaginary solution to the equation.
6x=-(19x+25)/(x+1)
(6x)(x+1)=-19x-25
6x^2+6x+19x+25=-19x-25+19x+25
6x^2+25x+25=0
:
6x^2+10x+15x+25=0
(6x^2+10x)+(15x+25)=0
2x(3x+5)+5(3x+5)=0
(2x+5)(3x+5)=0
2x+5=0 and 3x+5=0
2x=-5 and 3x=-5
x=-5/2 and x=-5/3
You're right!!!
Keep up the good work!
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Hi CH lam,
A church spire 82 metres high casts a shadow 62 metres long. Find the angle of elevation of the sun at that moment.
:
Draw a right triangle, the spire (I'm not sure what that is, I am assuming that it goes all the way down to the ground) is the side going up and down opposite the angle with the ground, the shadow is the side ajacent to the angle.
:
Opposite side=82m
ajacent side=62m
angle of elevation=x
:
tan(x)=opposite/ajacent


x=tan^(-1)(41/31)<--inverse tan or arctan
x=52.907 degrees
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Hi CH Lam,
A ladder 6.54 metres long leans against a wall, touching a window sill, and makes an angle of 62° with the ground. Find the height of the window sill above the ground. How far is the foot of the ladder from the foot of the wall ?
:
Draw a right triangle, the ladder is the hypotenuse, the wall is the side opposite the angle with the ground, the distance frome the wall is the side adjacent to the angle with the ground.
:
angle with the ground=62
height of the wall=o
distance from the wall=a
The ladder is the hypotenuse=6.54 m
:
sin(x)=opposite/hypotenuse



The wall is about 5.774 m
:
cos(x)=adjacent/hypotenuse



The distance from the wall is 3.07 m.
:
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The current in ther lazy river moves at a rate of 4mph. Monica'sdinghy motors 6mi upstream in the same time it take to motor 12mi downstream. What is the speed of the dinghy?
The distance formula is d=rt. d=distance, r=rate, and t=time. Solve the formula for t.
--->
distance up stream=6
rate upstream is the rate of the boat (r)- rate of the current(4):r-4
distance downstream=12
rate downstream is the rate of the boat (r)+ rate of the current(4): r+4
The time is the same for both, so set the distance formula solved for time equal to each other.












The dingy was moving at 12 mph.
:
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I can't speak for everyone. I can speak for myself. I think someone is making money, because students keep telling me they are sending someone donations. Some tutors have a side paid online tutoring businesses, I've seen them advertise. I personally don't make any money from this. I taught high school math for 5 years. Now, I am a math instructional assistant for a community college full time. We don't have good online software for tutoring math, so I tell my students that I monitor this site from time to time if they need help from home or work. I monitor the site while I'm in my office and have no students and when I've got nothing to do at home. I really like helping students acheive their goals. That's why I do what I do for a living. But this site gives me no monitary gain, I just like helping people with math. It's nice to be good at something and help others. I've been professionally tutoring for my college for 5 years, it's a lot more personally rewarding to personally tutor people than to teach a whole class.
Besides, math is fun!!!

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Instructions: solve each system by the eliminsation method.
What is the solution for x-y=4
x-y=-3
Don't freak out, but this one has no solution. If you try to eliminate either variable both go and you get 0=7 or 0=-7. Graphically, these lines are parallel and will never intersect.
Multiply the top equation by -1 and add them together.
-1(x-y)=-1(4)--->-x+y=-4
:
-x+y=-4
x-y=-3
_______
0+0=-7
0=-7 No solution.
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if sally can paint a house in 4 hrs, and john can paint the asame house in 6 hrs, how long will it take for both of them to apint the house together?
Let sally's rate be x/4
Let john's be x/6
:






x=2.4 hrs
or x=2hr+.4(60)min=2hrs and 24 min
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An airplane covered 25 miles of its route while decreasing its altitude by 19,000 feet. Find the slope of the airplane's line of descent. Round to the nearest hundredth. [Hint: 1 mi = 5280 feet.]
The plane covered horizontally.
Slope=rise/run or vertical/horizontal
slope=-19,000/132,000=-.14393939...=-.14
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Find the slope and the y-intercept. y = –9x + 8
The slope intercept form of a line is , m=slope, and (0,b) is the y-intercept.
the slope m=-9
the y-intercept is (0,b)=(0,8), some books and teachers let you say 8.
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on a true/false test a student has no idea what the answers are to 3 of the questions so he guesses. What is the probability that he will guess correctly on all 3 questions? at least 2 times?
There's 1 write answer out of two choices for each question. Therefore there's a probability of guessing all three questions right.
Percetagewise that's 12.5%. Note that that's the same probability of getting them all wrong (think about it.).
At least two right is 2 out of three and 3 out of three
There are three ways to get 2 right (wrr),(rwr),(wrr) and one way to get them all right (rrr)
++
ways to get 2 right
ways to get 3 right
ways to get 2 or 3 right. Note that this is the same probability of getting less than 2 right. (think about it.)
You can use binomial probability for this, but it would only be easier to do for a problem this small if you have a capable calculator. If you have a TI-83 or 84. Just hit, binomialpdf(3,.5,2)+binomialpdf(3,.5,3)=.5 or 1/2.
:
The formula for binomial probability is:

P(x)=probability of x
n=number of trials
x=number of successes
p=probabillty of success
q=1-p=probabilty of failure
For the probability of 2 correct answers:
n=3, x=2, p=1/2=.5, q=1-1/2=.5
For the probabilty of 3 correct answers:
n=3, x=3, p=.5, q=.5












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Find the slope of any line parallel to the line through points (19, 11) and (9, 9).
The slope through these two points can be found with the slope formula: , m=slope
(x1,y1)=(19,11) and (x2,y2)=(9,9)



Parallel lines have the same slopes, so any line parallel to the line made by these two points also has a slope of
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Find the slope of any line perpendicular to the line through points (8, 17) and (9, 9).
The slope between these two points can be found with the slope formual:
, m=slope, (x1,y1) and (x2,y2) are the given points.
(x1,y1)=(8,17) and (x2,y2)=(9,9)



Perpendicular lines have slopes that are negative reciprocals of each other. (That mean that they're opposite signs and upsidedown.)
The slope of any perpendicular line through the line created by those two points is:
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Rewrite the equation 2x – 3y = –6 as a function of x
Solve for y.
-2x+2x-3y=-2x-6
0-3y=-2x-6
-3y=-2x-6


Some books want you to also replace y with f(x)

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Given f(x) = x^2 – 3x + 9, find f(0).
Substitute 0 in where the x's are and simplify:



This is a special point called the y-intercept of the graph of the equation.
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Write the equation of the line that passes through point (–4, 19) with a slope of –6.
There are a couple of ways to do this, but whenever I'm given a slope and a point, I like to use the point slope formula , m=slope, (x1,y1)=given point.
m=-6, (x1,y1)=(-4,19)




<--This is the slope intercept form of a line.
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f(x)=(x^2-2x+8)/(x+4)
You didn't give us instructions.
x cannot = -4 has to be stipulated, because the original function is undefined when it's denominator equals 0
The domain therefore is {x|x cannot=-4}<--set builder notation
There is a vertical asymptote at x=-4
The degree of the numerator is higher than the denominator, so there is no horizontal asymptote.
The degree of the numerator is exactly 1 more than the denominator, so there is a slant asymtote you can get through long division of y=x-6.
If you plot a few points on both sides of x=-4 you get this graph.

If you had something else in mind, let me know.
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I can't seem to find the function rule for this set of data:
x y
-2 -2
0 1
2 4
4 7
If you plot the points and connect them you will see that they make a perfect line like this.

You were given the y-intercept (0,1), all we need is a slope and we can form the function.
The slope formula , m is the slope, (x1,y1) and (x2,y2) are two points on the line.
(x1,y1)=(-2,-2) and (x2,y2)=(0,1)



The slope intercept form of a line is , m=slope, and (0,b)=y-intercept.
Your m=3/2 and (0,b)=(0,1)
y=(3/2)x+1
You can replace the y with f(x) if you need function notation.
f(x)=(3/2)x+1
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Graph Indicate all intercepts and asymptotes.
The horizontal asymptote is y=0, because the degree of the numerator is less the the degree of the denominator.
:
The vertical asymptotes are when the denominator=0:

(x-5)(x+3)=0
x=5 and x=-3 are vertical asymptotes.
:
The x-intercept happens when the numerator =0.
4x+8=0
4x=-8
4x/4=-8/4
x=-2 The x-intercept is (-2,0)
:
The y-intercept happens when x=0.


The y-intercept (0,-8/15)
:

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Solve the following equations. Leave radicals in radical form(do not calculate there approximate values).
a.
Replace the middle term with two numbers that multiply to give you 4x^2*-10=-40x^2, but add to give you 3x. 8x*-5x=-40x^2, 8x+-5x=3x.
: Factor by grouping, group the first two terms and the last two.
Factor the GCF out of each parentheses
Factor out x+2
Set each parentheses equal to 0 and solve for x.
4x-5=0 and x+2=0
4x=5 and x=-2
4x/4=5/4 and x=-2
x=5/4 and x=-2
:
b. 2x^2=3x Set equal to 0
Factor out the GCF.
x(2x-3)=0
x=0 and 2x-3=0
x=0 and 2x=3
x=0 and 2x/2=3/2
x=0 and x=3/2
:
c. x^3-4x^2-x+4=0 Factor by grouping


x^2-1 is the difference of squares
(x+1)(x-1)(x-4)=0
x+1=0 and x-1=0 and x-4=0
x=-1 and x=1 and x=4
:
d. x^4-10x^2+9=0

(x+3)(x-3)(x+1)(x-1)=0
x+3=0 and x-3=0 and x+1=0 and x-1=0
x=-3 and x=3 and x=-1 and x=1
:
e. x^3-3x^2+x-3=0



(x^2+1 is prime, so if you need only real numbers you can ignore it, otherwise it's solution is +/-i)
x-3=0
x=3
:
f. 3x^2-5x+1=0
a=3 b=-5 and c=1
The quadratic formula is



:
An abreviation for radical is "sqrt(whatever you want)" like the square root of x+5 is sqrt(x+5)
:
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how can i solve the system by graphing.
E1: 2x + y = 4
E2: x + y = 3
:
For E1: let x=0 and solve for y.
2(0)+y=4
0+y=4
y=4 Plot (0,4)
Now let y=0 and solve for x.
2x+0=4
2x=4
2x/2=4/2
x=2 Plot (2,0)
Connect the points and you have a line.

For E2: let x=0 and solve for y.
0+y=3
y=3 Plot(0,3)
Let y=0 and solve for x.
x+0=3
x=3 Plot (3,0)
Connect the points and you have another line:

They connect at (1,2)<--this is the solution of the system.
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how do i write the equation of the line with slope 4 and y-intercept (0, –5). Then graph the line.
The slope intercept form of a line is , m=slope, and (0,b)=y-intercept.
Your m=4 and b=-5

To graph the line put a point on the y axis at -5.
From that point use the slope=rise/run=4/1. Go up 4 and right 1 plot another point. (This should be (1,-1) if you did it right.)
Connect the points and you have a line.

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Find all real solutions to
(a)
Use the substitution method.

Let
Factor
(P+2)(P-1)=0
P+2=0 and P-1=0
P=-2 and P=1
But we want to know what x is, so let
and
and
and
and
You have to use the quadratic equation to solve these:

when a=1, b=1, c=-4


You have to check these to see if they are valid or extraneous. I checked these by plugging them into my calculator and the were both extraneous.
when a=1, b=1, and c=-1

You have to check these to see if they are valid or extraneous. I checked these with my calculator and they were both valid.
If you have a TI-83 or 84, I can give you instructions on checking for extraneous solutions if you ask.
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For the following, letf(x)=x+5; let g(x)=x^2-2; let h(x)=-3x
for h(f(4)) I set 4=x, the value of f(4) then being 9; the expression h(9) would then be -27???right???RIGHT!!!!
:
for g(x+3), I set (x)=x+3 and then the value of g(x+3)
g(x+3)=(x+3)^2-2
g(x+3)=(x+3)(x+3)-2
g(x+3)=x^2+3x+3x+9-2
g(x+3)=x^2+6x+7
:
1. Same parameters (f-g)(x)
(f-g)(x)=(x+5)-(x^2-2)
(f-g)(x)=x+5-x^2+2
(f-g)(x)=-x^2+x+7
:
2. h(2x^2+4x)=-3(2x^2+4x)
h(2x^2+4x)=-6x^2-12x
:
I had trouble interpretting this one. From what I can tell you are putting ^ before AND after your exponents. (You only need it before, and if you have a fractional exponent, put it in parentheses.) If this isn't what you meant, let me know.
3. 5x^(3/2)-25=0
5x^(3/2)-25+25=0+25
5x^(3/2)=25





:
Check by substitution to see if it's true.




0=0 we're right!!!
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given a line containing (1,3),(2,4),(3,5)determine the slope-intercept from of the equation and give one additional point on this line and graph the function.
We need a slope and a point to make the equation of a line.
We don't have the slope, but we can find it using the slope formula:, m=slope, (x1,y1) and (x2,y2) are given points.
Let (x1,y1)=(1,3) and (x2,y2)=(2,4)


(I checked all the points, you can use any of them to get m=1)
We can now use the point slope equation to get the equation:, m=slope, and (x1,y1) is a given point.
m=1 and (x1,y1)=(1,3)
y-3=1(x-1)
y-3=x-1
y-3+3=x-1+3
y=x+2 <--this is slope intercept form , m=slope and b=y-intercept.
:
You can find another point by letting x= a number, like 0 and solve for y.
y=0+2
y=2
Plot all these points and connect them and you should have this line:

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Find the numerical value of the expression sin 30degrees + cos 60degrees.
The answer choices are:
1.
2. 1/4
3. square root of 3
4. square root of 3 + 1/2
There are two special right triangles that you have to memorize the values of. One is the 30, 60, 90 degree triangle. Draw the triangle, label the side opposite the 30 degree angle 1, the side opposite the 60 degree angle is , and the side opposite the 90 degree angle (the hypotenuse) is 2.
sin(30)=opposite/hypotenuse=1/2
cos(60)=adjacent/hypotenuse=1/2
sin(30)+cos(60)=1/2+1/2=2/2=1
The other special right triangle you have to memroize is the 45,45,90 triangle. On that one the hypotenuse is , and the other two sides are 1.
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given that.
f: 3x^2-4x+1
g: 2x-1
calculate fg(10)
This means evaluate the function of f times the function of g when x=10.
:
fg(x)=(3x^2-4x+1)(2x-1)
fg(10)=(3(10)^2-4(10)+1)(2(10)-1)
fg(10)=(3(100)-40+1)(20-1)
fg(10)=(300-40+1)(19)
fg(10)=(261)(19)
fg(10)=4959
:
You could have also just found f(10) and g(10) and multiplied them if you're intimidated by the size of the equation.
f(10)=261
g(10)=19
fg(10)=261*19=4959
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The formula you need to remember is:
Factor completely


Use the formula letting a=4x and b=3


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The cost of producing a number of items x is given by C = mx + b, in which b is the fixed cost and m is the variable cost (the cost of producing one more item).
(a) If the fixed cost is $40 and the variable cost is $10, write the cost equation.
b=40, m=10

(b) Graph the cost equation.

(c) The revenue generated from the sale of x items is given by R = 50x. Graph the revenue equation on the same set of axes as the cost equation.

(d) How many items must be produced for the revenue to equal the cost (the break-even point)? The lines intersect at (1,50), therefore the number of items that must be produced to break even is x=1.
You could have found this algebraically by letting C=R
10x+40=50x
10x-10x+40=50x-10x
40=40x
40/40=40x/40
1=x
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When a ball is thrown vertically upward, its height (h, in feet) above the ground after t seconds is described by the mathematical formula h = - 16t2 + 96t + 80. At what times is the ball 103 feet above the ground?



Use the quadratic formula:
a=16, b=-96, c=23



and
and
and
x=.25 s and x=5.75 s
One the way up, the ball hits 103 ft in .25 s, on the way back down it hits it again in 5.75 s.
Since this came out to rational answers this could have been factored. When the numbers are more difficult to deal with the quadratic formula is easier to use than factoring. Here's what it looks like factored.

4t-1=0 and 4t-23=0
4t=1 and 4t=23
t=1/4 s and t=23/4 s
Happy Calculating!!!