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Systems of equations with substitution: shelves

Sal solves a system of equations to figure out how long a couple of different shelves will be. Created by Sal Khan and Monterey Institute for Technology and Education.

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• Jenny mixes a 30% saline solution with a 50% saline solution to make 800 milliliters of a 45% saline solution. How many milliliters of each solution does she use?
• By selling 12 dozen pencils, Alice lost an amount equal to the selling price of 6 pencils. Find the lost percent.
• Ok. So how many pencils do you have in total? If a dozen is 12, and you have 12 dozens, you can multiply it out and reach 144. Now she lost a certain percentage of money that is equal to 6 pencils. Now, you just find out how much 6/144 is, but with a denominator of 100. If you simplify this fraction (6/144), you will get 1/24. So, you would end up with 0.04166666 (you get the idea, it's infinite 6's), and you could say that it is approximately 4.16666 %. Hope this helped!
• Is there a quicker way to do this?
• Step 1: Write down what you know:
1. We know there are 2 investments
2. We know that the total dollars invested in those investments = 5,800
3. We know that one investment, represented by the letter x earns 3.5% interest
4. We know that the other investment is represented by the letter y, and earns 5.5% interest
5. We know that they are asking us to solve for the total dollars invested in investment y

Step 2: Come up with equations with the information that was given:
Dollars invested in first investment = x
Dollars invested in second investment = y
Equation 1: x + y = total dollars invested = 5,800
Equation 2: 0.035x + 0.055y = 283,
I find it easier working with whole numbers, so I will multiplying both sides of
equation 2 by 1000 to get 35x + 55y = 283,000

Step 3: Isolate one of the variables, then use substitution to solve:
We are looking for y, so we need to substitute x for y, using the first equation, we get
x = 5800 - y, substituting that value for x into the second equation, we get
35(5800 - y) + 55y = 283,000, or 203,000 -35y + 55y = 283,000, or
20y = 80,000, y = \$4000. So, \$4000 was invested at 5.5%
and \$1800 (5,800 - 4,000) was invested at 3.5%.
4000 * 0.055 + 1800 * 0.035 = 283.
(1 vote)
• This is for those of you that need a challenge :D Here is a systems of equations word problem that I found online.

The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number.
• The digits of a two digit number must be integers so this greatly reduces the number of possibilities. First, what pairs add up to 7?
1 + 6
2 + 5
3 + 4
And the reverses of these. Immediately we can see that the number is either 16, 25, or 34, and cannot be 61, 52, or 43 because the question says when the digits are reversed the number INCREASES so the original number must have a smaller tens digit than units digit.
61 is 45 greater than 16.
52 is 27 greater than 25.
So the number is 25.
• a small plane leave san jose airport and flies to los angeles at 240 miles per hour. a jet plane leaves san jose airport 30 minutes later to follow small plane at 360 miles per hour. how long does it take the jet to overtake to the small plane
• I got a different answer being one hour. So let us say that y is the distance from the airport and x the amount of hours so for the small plane the equation would be 240x + 240 * 0.5 = y (i got 0.5 because 30 minutes = 0.5 hours) and the equation for the jet being 360x = y. So then you set them equal so 360x = 240x +240 * 0.5. simplify it 360x = 240x + 120. get x on one side 120x = 120. and get x alone. x=1 so just one hour to pass it.
(1 vote)
• why are there no practice problems for the word problem
• You can suggest for Khan Academy to add practice word problems for this topic in the Help Center (You can find it in the blue area at the bottom of this website under "contact").
• help me solve this problem two linear problem 3x+y =2 3x+2y=7
• Joann,
When you have two equations like
3x+y =2 and
3x+2y=7
And you want to use the substitution method, you first need to isolate either variable by itself on one side of one of the equations.
In this case, the easiest way is to isolate the y on the right in the first equation.
3x+y =2
Subtract 3x from both sides.
3x-3x+y = 2-3x
The 3x-3x becomes 0 and disappears
y=2-3x
Now you can use that information and just
substitute (2-3x) for the y in the second equation
3x+2y=7
Substitute (2-3x) for the y
3x + 2(2-3x) = 7
Now you can find the solution for x

Then you can use that answer for x by putting it in either equation for the x and solve that equation for y

Then you can check you answer by putting the answers for both x and y into the other equation and see that it balances correctly.

I hope that helps make it click for you.
• Please break this down for me, I am stuck.
You invest a total of \$5800 in two investments earning 3.5% and 5.5% simple interest. Your goal is to have a total annual interest income of \$283. Write a system of linear equations that represents this situation where x represents the amount invested in the 3.5% fund and y represents the amount invested in the 5.5% fund. Solve this system of to determine the smallest amout that you can invest at 5.5% in order to meet your objective.
• Step 1: Write down what you know:
1. We know there are 2 investments
2. We know that the total dollars invested in those investments = 5,800
3. We know that one investment, represented by the letter x earns 3.5% interest
4. We know that the other investment is represented by the letter y, and earns 5.5% interest
5. We know that they are asking us to solve for the total dollars invested in investment y

Step 2: Come up with equations with the information that was given:
Dollars invested in first investment = x
Dollars invested in second investment = y
Equation 1: x + y = total dollars invested = 5,800
Equation 2: 0.035x + 0.055y = 283,
I find it easier working with whole numbers, so I will multiplying both sides of
equation 2 by 1000 to get 35x + 55y = 283,000

Step 3: Isolate one of the variables, then use substitution to solve:
We are looking for y, so we need to substitute x for y, using the first equation, we get
x = 5800 - y, substituting that value for x into the second equation, we get
35(5800 - y) + 55y = 283,000, or 203,000 -35y + 55y = 283,000, or
20y = 80,000, y = \$4000. So, \$4000 was invested at 5.5%
and \$1800 (5,800 - 4,000) was invested at 3.5%.
4000 * 0.055 + 1800 * 0.035 = 283. It checks.
Hope that helps and good luck with your studies!
• Is there any practice for this topic?
• I asked this question on a diffferent video but I need this question answered as fast as possible!!
"A certain number when added to 36 gives the same result as adding 6 to this number and doubling the result".
(1 vote)
• "A certain number..." -> x

"...when added to 36..." -> x + 36

"...gives the same result as..." -> x + 36 =

"...adding 6 to this number..." -> x + 36 = x + 6

"...and doubling the result." -> x + 36 = 2(x + 6)

Then...

x + 36 = 2x + 12
x = 24