Solve each problem. See Examples 5 and 9. A cashier has a tota...

Question

Solve each problem. See Examples 5 and 9. Solve the system of equations (4), (5), and (6) from Example 9.

$25 x + 40 y + 20 z = 2200$ (4)

$4 x + 2 y + 3 z = 280$ (5)

$3 x + 2 y + z = 180$ (6)

Accepted Answer

Hello everybody. I I today, today we're going to be looking at this question that states determine the quantity of each type of ball purchased by a sports teacher who bought a total of 40 balls. So 40 balls is our total here. The number of basketballs exceeds the number of footballs by eight. Given that the cost of a tennis ball is $4. A basketball is $5 and a football is $7. The total cost of the purchase was $184. So the extras provided here are a number of tennis balls is 12, number of basketballs is 24 number of footballs is four, B, number of tennis balls is four, number of basketballs is 12 and footballs is C. We have 24 tennis balls, 12 basketballs and four footballs and D. We have 24 tennis balls, four basketballs and 12 footballs. So when it comes to a question like this, I see that they provide me with different amounts and different costs and such. And one of the first things I want to do here is assign variables to each of our components. So in this case, I'm gonna say that X is equal to the footballs or the amount of footballs. Why will be basketballs? N Z will be tennis balls. So once we assign a variable to each of our components, we can go ahead and start solving our question. So we're set an equation here. We know that X plus Y plus Z equals 40 because they told us that the um total amount of balls is 40. So if we add, we add up all of our balls which are X X, Y and Z, we'll have an answer of 40. Now, they also told us that the number of basketballs exceeds the number of footballs by eight, which means Y is equal to X plus eight. And now we can plug that into our equation. So we'll have X plus first, we had Y, but now we know that's gonna be X plus eight. So we'll have X plus X plus eight plus Z equals 40. And now we can simplify this a bit better. So we know that X plus X will be two X. So we have two X plus eight plus Z equals 40. And if we move, move the eight over from the left hand side to the right hand side by subtracting. And we'll also move the two X over from the right hand side to the left hand side to subtract to solve for Z, we'll have that Z is equal to 32 minus two X. So now we have an equation for Y and an equation for Z and an equation for all three variables added up. So now we can come up with a cost equation. So they told us the price that each ball costs. So for example, the footballs are $7 each. So we can have $7 or seven multiplied by X. That will give us the total cost of that amount. So we have seven X plus and then we'll have five Y. So seven X plus five Y because five is the cost of a basketball plus four Z because $4 is the cost of a tennis ball that will equal 100 and 84 because 100 and 84 was the total cost of the whole thing. So now we do our plugging in of what we know already. We'll have that seven X plus five multiplying and open parentheses, X plus eight closed parentheses plus four, multiplying an open parentheses. 32 minus two, X closed primes are seized equals 184. And all we did there was plug in what we know to be the Y which is X plus eight and what we know to be the Z which is 32 minus two X and now we can distribute everything. So we'll have seven X plus five, multiplied by X is five X and five multiplied by eight is 40. So five, so seven X plus five X plus 40 plus four, multiplied by 32 will be and then four multiplied by negative two X will be minus eight X and that will be equal to 184. And now, if we just simplify this, we'll have seven X plus five. X is 12, X minus eight, X is four X and 40 plus 128 will be plus and that will be equal to 184. And if we subtract or we move the 168 over from the left hand side to the right hand side by subtracting, we'll have that four, X is equal to 16. And now we can just divide both sides by four to get that X is equal to four footballs. And that will be the first answer for this question. So now we can continue to do some plugging in with that information. Remember our equation for Y or our equation for basketballs was Y equals X plus eight. And now we have the X value so we can plug that in. So to have Y equals four plus A, which means that Y equals 12 basketballs. And that'll be the second answer to our question. And finally, we can plug into our Z value or our tennis balls. Remember we had Z equals minus two X and we now have the X value. So we will have Z equals minus two, multiplied by four, which means that Z equals 32 minus eight, giving us an answer of Z equals tennis balls. And that'll be our third answer for this question. So we have a number of tennis balls at 24 number of basketballs at 12 and the number of footballs at four which all match up with answer choice C for this question. So I hope that was helpful and until next time.

Solve each problem. See Examples 5 and 9. A cashier has a total of 30 bills, made up of ones, fives, and twenties. The number of twenties is 9 more than the number of ones. The total value of the money is \$351. How many of each denomination of bill are there?

Key Concepts

Setting Up Variables for Word Problems

Formulating Systems of Linear Equations

Solving Systems of Equations

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