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Q:The sum of two numbers is 38 and their difference is 26 more than the smaller one. What are the two numbers?
A: You will want to translate the english version of this problem into algebra. Start with the first part of the first sentence: The sum of two numbers is 38.
We know that SUM is ADDITION, so this meas two numbers added together equal (IS) 38. So if the smaller number is x and the larger is y, then
x + y = 38.
Now, the second part of the sentence, their difference is 26 more than the smaller one. The DIFFERENCE means subtraction. We subtract the two unknowns, y - x since x was the smaller. This is equal to 26 more than the smaller. MORE THAN means addition (x + 26). So now we can build a second equation,
y - x = x + 26.
Depending on what level of mathematics you are in you can solve using two variables or substituting 38-x for y in the second equation and then using just one equation and one variable.
x + y = 38 & y - x = x + 26:list both equations
y = 38 - x & y = 2x + 26: solve for y
38 - x = 2x + 26:substitute for y since both equations are equal to y
12 = 3x: subtract 26 from both sides and add x to both sides
4 = x:divide by 3
y = 38 - x :go back to your first equation
y = 38 - 4 :substitute x=4
y = 34 :subtracct
Double check: y - x = x + 26
y = 34, x = 4
34 - 4 ?=? 4 + 26
30 = 30
So the correct solution are the numbers 4 and 34 just as you had found.

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We are looking for the algebraic expression for six less than half a number
1) are there variables- yes "a number" is an unknown amount- substitute in x
now we are looking for the algebraic expression for six less than half x
2) working from the variable out the first word we get to is half- half is equal to .5 or 1/2.
now we are looking for the algebraic expession for six less than .5x
3) work from the variable out again and we get to less than. Less than means subtraction, but order matters wth subtraction. So which side is being subtracted from the other?? Whenever you see less than you can switch the order the two sides are in when you replace the words with the symbols. (Think: If I had six candy bars less than my brother would I be happy or sad? The answer here is sad since this means my number of candy bars is smaller than my brothers by six)
now we have which is our solution

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To make an open box we can cut squares out of the corners and fold the sides up. If x is the length of the side we are cutting then when we fold that side up, x will be the height (take a minute to think about what this looks like when you cut out the box length x and then fold that flap up. the x inch piece you cut is the amount you are folding up so that is the height.)
So x is the height. We are looking for the volume so we will also need the length and width since the formula for volume is the product (*) of length, width and height.
We cut off x inches off the length and width on each side (again think about what this looks like). We are taking 2x off each measurement (one x from each end). So now instead of a length of 8 inches, our length is 8-2x inches.
Similarly, our width of 10 inches was cut down by 2x also to give us 10-2x inches.
Now we have in terms of x the length, width and height. The volume is the product of these three, so substitute them in. Let V(x) be the volume.
V(x)=length* width *height= (8-2x)*(10-2x)*(x)
We'll simplify for fun- FOIL the first two


Distribute x over each term

Note that the volume is going to be in cubic inches (inch*inch*inch)=(inch^3)

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The formula for Area is Area= Length * Width for both a rectangle or a square. The sides of a square are both equal. Let's call the length of each side of the square x. The measurements of the rectangle are listed as 4 feet narrower and the other side is 5 feet longer. This means their measurements are x-4 (the narrower side since narrower means smaller) and x+5 (since longer would be more).
The area of the square is x*x or x^2. The area of the rectangle is (x-4)*(x+5). These two areas are equal by the first sentence. So x^2=(x-4)(x+5). We use FOIL (first, outer, inner, last) to multiply out the right side.
x^2=x^2+5x-4x-20. Simplify terms of the same degree (same exponent value)
x^2=x^2+x-20.
Next we set the left side equal to zero by subtracting x^2 from both sides.
0=x-20
Now solve for x.
x=20.
Remember that x was the length of the sides of the square so our value has units of feet (from the second sentence).
The solution asks for the area of each room so we will need to determine this by going back to the area formula. Since the areas are equal we don't need to calculate both, but we can to double check our work (what's better then knowing you did everything right?!)
Area=length * width. The length and with of the square are both 20.
Area of the square= 20*20=400. Now don't forget units! The solution is 400 square feet (or ft^2)
Double check:
Area of the rectangle=Length* width. The two sides are (20-4)=16 and (20+5)=25 (remember x was 20) 16*25=400. We did everything correctly!!!