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Start by letting R = Robert's present age and F = Father's present age.
From the problem you can write:
1) "Robert (R) is half as old as his father (F)"
2) "Twelve years ago, he was 1/3 as old as his father was then"
So now you have a system of equations with two unknowns (R and F) and you can use any of the accepted methods for solving these for R and F.
You can simplify these equations by clearing the fractions.
1) Multiply through by 2.
1a)
2) Multiply through by 3.
2a) Now substitute the F from the simplified equation 1a) into the simplified equation 2a).
Simplify and solve for R by subtracting 2R from both sides.
Add 36 to both sides.
This is Robert's present age.


This is the father's present age.
Check:


OK, and...



OK

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Write the equation, in slope-intercept form, of the line that passes through the points (7, 6) and (1, -12)
You can find the slope, m, of the line using:
where: , , , and



Now you can start the equation as:
Replace the m with the slope just calculated: m = 3
Now we need to find the value of b. You can do this by substituting into this equation, the x- and y-coordinates of either one of the two given points. Let's use the first point (7, 6)
Substitute x = 7 and y = 6
Simplify and solve for b.
Subtract 21 from both sides.
Now you can write the final equation:

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Factor by grouping:
Group the terms as shown below:
Now factor 7x from the 1st group and 4 from the 2nd group.
Finally, factor the common factor (x+y)
...and there it is!

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Solve by addition: Note, the idea behind solving systems of equations is to eliminate one (or more) of the variables using any of the methods of addition, subtraction, substitution, etc.
2x-4y = 7
4x+2y = 9 To eliminate one the variables by addition, you first must multiply one or the other of the two equations by an integer so that you end up with the same number of one of the variable (either x or y) in both equations.
Since you are required to use addition, let's multiply the first equation by -2
-2(2x-4y = 7) to get:
-4x +8y = -14 Now add this equation to the second equation.
4x+2y = 9
----------
Notice that we have eliminated the x's Now divide both sides by 10.
Simplify.

Now we substitute this value of y into either one of the two original equations then solve it for x. Let's use the second equation:
Substitute
Simplify.
Add 1 to both sides.
Divide both sides by 4.
Simplify.

The solution is: (2.5, -1/2)
Let's see what the graph of these two equations looks like:

The solution is where the lines intersect.

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You can start by assigning variables (letters) to the unknown quantities.
Let C = the number of cows and B = the number of bulls.
From the story, you can write:
C = 20B "...bought 20 times as many cows as bulls."
The cost of each cow is $1,000 and this can be represented as ($1,000)*C
The cost of each bull is $2,000 and this can be represented as ($2,000)*B
The total cost (sum) of these two purchases is $220,000
Now we can write the equation to solve the problem:
($1,000)C + ($2,000)B = $220,000 but we can replace the C here with its equivalent C = 20B toget:
($1,000)(20B) + ($2,000)B = $220,000 Simplify and solve for B
($20,000)B + ($2,000)B = $220,000 Combine like-terms.
($22,000)B = $220,000 Divide both sides by $22,000 to get:
B = 10 The rancher bought 10 bulls.
C = 20B
C = 20(10)
C = 200 The rancher bought 200 cows.
Check:
($1,000)(200) + ($2,000)(10) = $200,000 + $20,000 = $220,000

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You can get the answer to this problem by subtracting 1/5, 1/8, and 1/10 from 1 because 1 represents her entire portfolio, thus:
The LCD here is 40, so:
=
The remainder of her portfolio, is invested in Municipal bonds.
To find the equivalent percentage, you would divide the numerator (23) by the denominator(40) and multiply the result by 100 to get the percent.
Multiply by 100 and add the % sign.
= 57.5%

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Start by letting H = the price of a hamburger, F = the price of french fries, and C = the price of a Coca Cola.
From the problem description, you can write:
H = 2F "The price of a hamburger (H) is (=) twice the price of french fries (2F)"
C = F+$0.40 "The price of a Coca Cola (C) is (=) 40 cents more than the price of french fries (F+$0.40)"
Now the sum of these three items can be had for less than $4.00 (<$4.00), so we can write the inequality:
H+F+C < $4.00 Now we substitute H = 2F and C = F+$0.40 to get:
2F+F+F+$0.40 < $4.00 Combining like-terms, we get:
4F+$0.40 < $4.00 Subtracting $0.40 from both sides of the inequality, we have:
4F < $3.60 Dividing both sides by 4, we end up with:
F < $0.90 or, in words, the price of french fries is no more than $0.90

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I answered this question yesterday (10/20/07) at 19:45:04
See if you can find it and, if you still have questions about my answers, please e-mail me and I will do my best to resolve them.
It was question # 105311
My answer was # 76634
"Earlsdon"

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Solve by addtion:
2x-y = 1
-2x+3y = 5 Add the two equations and the x's dissapear leaving you with:
----------
2y = 6 Divide both sides by 2.
y = 3 Now substitute this value of y into either one of the two original equations and solve it for x.
2x-y = 1
2x-3 = 1 Add 3 to both sides.
2x = 4 Divide both sides by 2.
x = 2
Check: Substitute x = 2 and y = 3 into the two given equations.
1st equation:
2x-y = 1
2(2)-3 = 1
4-3 = 1
1 = 1
2nd equation:
-2x+3y = 5
-2(2)+3(3) = 5
-4+9 = 5
5 = 5

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Let x = the amount invested at 3.5%, then ($25000-x) = the amount invested at 4%.
The dividend earned on the first amount is 3.5% of x and this can be written as: 0.035x.
The dividend earned on the second amount is 4% of ($25000-x) and this can be written as 0.04($25000-x)
The sum of these two dividend amounts is given as $900
We have enough now to write the necessary equation to solve for x.
0.035x + 0.04(25000-x) = 900 Simplify and solve for x.
0.035x+1000-0.04x = 900 Combine like-terms.
-0.005x+1000 = 900 Subtract 1000 from both sides.
-0.005x = -100 Divide both sides by -0.005
x = 20,000
$20,000 was invested at 3.5%
$5,000 was invested at 4%
Check:
$20,000(0.035)+$5,000(0.04) = $700+$200 = $900 Total dividends earned.

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Your answer is correct and you can always check by multiplying the two factors.
The only thing is, we usually write the answer like this:

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a)
Let C = the royalties for Cats and F = the royalties for Fame. P = the number of performances.
C = $1250 + $325P
F = $1400 + $250P
b)Cats is the red line and Fame is the green line. The horizontal axis is the number of performances (P) and the vertical axis is the total amount of royalties for each of the two musicals..

c)
To find the value of P when the royalties are the same for both plays, set C = F
$1250+$325P = $1400 + $250P Subtract $1250 from both sides.
$325P = $150+$250P Now subtract $250P from both sides.
$75P = $150 Divide both sides by $75
P = 2
The royalties are the same for second performance.
d) One reason is that the creators are guaranteed to get at least $1250 for Cats and $1400 for Fame even if there are no performances.

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One way to do these is to substitute the x- and y-values from the given points into the equation, thus giving you a system of equations with two unknowns (a and k). Then solve the system of equations for the two unknowns.
Substitute x = 1 and y = 11 from the first point (1, 11) to get the first equation.
Simplify.

Second equation:
Substitute the x- and y-values from the second point (2, -19) to get the second equation:
Simplify.

So now you have the two equatios:
1) and
2) You can solve these by any of the accepted methods for solving systems of equations. Let's use elimination by subtracting equation 1) from equation 2) and solving for a.
Divide both sides by 5.
Now substitute this value of a into either equation 1) or equation 2) and solve for k. Let's use equation 1)
Substitute a = -6

Add 24 to both sides.

Check: Into the original equation, substitute a = -6 and k = 35.

Now let's see if the two given points (1, 11) and (2, -19) satisfy this equation.



Ok for the first point.



Ok for the second point.

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Try this:
Standard form is;

Expand the parentheses by squaring.
Simplify.

...and there you have it!

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Let L = the length and W = the width.
From the problem description, you can write:
L = 5W+2 "The length (L) of the rectangle is (=) 2 more (+2) than 5 times its width (5W)"
A = 65 sq.cm. "The area of the rectangle is 65 sq.cm."
The formula for the area of a rectangle is:
A = L*W Make the appropriate substitutions.
Simplify and solve for W.
Subtract 65 from both sides.
Use the quadratic formula to solve for W.

Discard the negative solution.

The width is 3.41 cm

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Solve:
1)
Factor this trinomial.
Apply the zero products principle:
or
Subtract 10 from both sides., so...

Add 2 to both sides.

2)
Factor this trinomial.
Apply the zero products principle.
or
Subtract 10 from both sides.

Subtract 6 from both sides.

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Let x = the fixed fee and y = the hourly rate.
For the 2-hour call, you can write:
1) ...and for the 4.5-hour call, you can write:
2)
You need to solve this system of equations for x, the fixed fee.
Solve equation 1) for y and substitute into equation 2) then solve it for x.
1)
2) Simplify and solve for x. Multiply both sides by 2.





The fixed fee is $25.00
The hourly fee is:
Substitute x = 25.



The hourly fee is $16.00

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Well...
The general form of the equation that gives the height (h) of an object propelled upwards, as a function of time (t) is:
where: h is the height in feet, t is the time in seconds, g is the acceleration due to gravity (, is the initial velocity of the object, and is the initial height of the object.
If you wanted to graph such an equation (it would be a parabola opening downward), it would be written as:

The x in your equation is a replacement variable for the time, t.
The first term is negative because of the downward effect of gravity on the object.
Just for fun, let's see what your equation would look like when graphed:

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Solve the inequality:
1) or is it?
2)
1) Multiply both sides by (x+1)
1) Now multiply both sides by x.
Subtract x from both sides.
Finally divide both sides by 4.
This is the answer for 1) but, if it's 2)
2) Multiply both sides by x.
Simplify.
Subtract 5 from both sides.
This is the answer for 2)
Hint: Use parentheses to clarify the intent of the problem.
In your problem you could have written:
(5/x)+1 or 5/(x+1) to remove any ambiguity.

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Identify the terms, like-terms, coefficients, and constant terms of:
-4k-12+3k
The terms are the monomials separated by a + or a -, so they are:
-4k, -12, 3k Notice that we usually include the sign preceeding the term.
The like-terms are the terms (more than one of them) that have a common variable, in this case it's k, or if there are terms (more than one of them) that have no variable, then those together would be like- terms.
-4k, and 3k
The coefficients are the numbers just in front of the variables.
-4 and 3
Constant terms are those terms that have no variables attached.
-12 is the only constant term, again, notice the negative sign is included.
Simplify: (By the way, it is not an equation but an expression)
-4k-12+3k Combine the like-terms together.
(-4k+3k)-12 this becomes -k-12 when you combine -4k and +3k

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Solve for d:
First, simplify the right side.
Now factor the d from the numerator of the right side.
Now multiply both sides by the inverse of

I'm not sure I understood you equation completely. Is it supposed to read:
?
If so, then the answer would read:

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You are right on!
Think of the "per" as "divided by", so:
Population "per" square mile is population "divided by" square miles.
For Asia, you get:
4,028,000,000/17,400 = 231,494 people per square mile.( Rounded to nearest integer, since you normally don't have fractions of people)
For North America:
305,000,000/9,400 = 32,447 people per square mile.
I'll leave the comparison to you!

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I presume that you need to solve for w?
6(w-5)-3w = w+10 First, apply the distributive property to the left side.
6w-30-3w = w+10 Now combine the variable, w.
6w-3w-30 = w+10
3w-30 = w+10 Subtract w from both sides.
2w-30 = 10 Now add 30 to both sides.
2w = 40 Finally, divide both sides by 2.
w = 20
Check:
6(w-5)-3w = w+10 Substitute w = 20
6(20-5)-3(20)= 20+10 Simpllify.
6(15)-60 = 30
90-60 = 30
30 = 30 OK!

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First, let's find the angle, which we can call angle A, then we can find its complement (90-A) to answer the problem.
The supplement of A is 180-A and the complement of A is 90-A
From the problem description, we can write:
2(180-(90-A)) = (180-A)+60 Simplify this and solve for A
2(180-90+A) = 180+60-A
2(90+A) = 240-A
180+2A = 240-A Add A to both sides of the equation.
180+3A = 240 Now subtract 240 from both sides.
3A = 60 Finally, divide both sides by 3.
A = 20 degrees. Now we can find the complement of A (90-A) as required by the problem.
90-A = 90-20 = 70 degrees.
The solution is:
The complement of the angle is 70 degrees.

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Ok Jessica, here we go:
1. It will simplify things a bit if we first subtract the fractions on the left side.
Now multiply both sides by f.
Now multiply both sides by the inverse of . This is the same as dividing both sides by

2. Solve for b. First, multiply both sides by 2.
Now divide both sides by h.
Finally, subtract a from both sides.

3. Solve for Q (I hope I interpreted this correctly!) First, factor the Q.
Simplify the right side.
Now multiply both sides by the inverse of

Done!

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Solve by substitution:
1) 5x-2y = 5
2) y-5x = 3
You have a choice. You can solve this equation for y and substitute into equation 1) and then solve that for x...or...you can solve this equation for 5x and substitute into equation 1) and solve that for y.
Let's try the first choice, solve equation 2) for y.
2) y = 5x+3 Now substitute this into equation 1)
1) 5x-2(5x+3) = 5 Simplify.
5x-10x-6 = 5
-5x-6 = 5 Add 5 6 to both sides.
-5x = 11 Divide both sides by -5
x = -11/5 Now substitute this for x into either of the two original equations and solve for y. Let's use the second equation:
y-5x = 3 Substitute x = -11/5
y-5(-11/5) = 3 Simplify.
y+11 = 3 Subtract 11 from both sides.
y = - 8
The solution is: (-11/5, -8) Let's check the graph of the two lines represented by the two equations.
The solution will be seen as the point of intersection of the two lines.

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Solve:
10+3(x+2) = 31 First, subtract 10 from both sides.
3(x+2) = 21 Now divide both sides by 3.
x+2 = 7 Finally, subtract 2 from both sides.
x = 5
Check: Substitute x = 5.
10+3(5+2) = 31
10+3(7) = 31
10+21 = 31
31 = 31

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Your answer is absolutely correct!
0.9y-18 = 0.3y Subtract 0.3y from both sides.
0.6y-18 = 0 Add 18 to both sides.
0.6y = 18 Divide both sides by 0.6
y = 30

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Let the two consecutive even integers be x and x+2
From the problem description, you can write:
3(x+2) = x-10 Simplify and solve for x.
3x+6 = x-10 Subtract x from both sides.
2x+6 = -10 Subtract 6 from both sides.
2x = -16 Divide both sides by 2.
x = -8 and
x+2 = -6
The two integers are: -8 and -6

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Since Terry has already walked 3 of the 13 miles, you know that he has (13-3) miles to go.
Remember that: d = rt or distance = rate(speed) multiplied by time.
You can write this as:
t = d/r and in this problem, r = 4 mph and d = (13-3), so you can write the equation:
t = (13-3)miles/4mph This will tell you how long, in hours, it will take Terry to cover the remaining miles.