Solve each problem. See Examples 5 and 9. At the Berger ranch,...

Question

Solve each problem. See Examples 5 and 9. At the Berger ranch, 6 goats and 5 sheep sell for $305, while 2 goats and 9 sheep sell for $285. Find the cost of a single goat and of a single sheep.

Accepted Answer

Hello everybody. We're doing all right. Today, today we're going to be looking at this question that states it, the prices of a sack of dog food and a sack of cat food. If in a pet shop, seven sacks of dog food and six bags of cat food were sold for $627. While three sacks of dog food and eight sacks of cat food were sold for $589 with no discounts applied. Now, the answer choices provided are a one sack of dog food equals $29.01 sack of cat food equals $69. B dog food is $39 and cat food is $59. C dog food is $59 and cat food is $39. D 69 dog food is $69 and cat food is $29. No, with a question like this where they give me different informations about different variables going on. I want to assign a variable or a specific variable to each part of my problem. So in this case, I'll say that X is equal to price of one sack of dog food and Y is equal to price uh one sack of cat food. So now I can write two different formulas. So in this case, I'll have equation number one will be seven X plus six, Y equals 627. And where does that come from? Well, they told us that seven sacks of dog food and six bags of cat food were sold for $627. So if X is the price of dog food, we'll have seven bags multiplied by that price, X plus six bags multiplied by the price of cat food, which is why. And if you add those up, it'll equal the total price which is 627. And using that same logic, we can write a second equation for the other values they gave us, which would now be three X plus eight Y equals 589. And that is for when they told us that three sacks of dog food. So three eggs and eight sacks of cat food. Eight Y add those up will give us total dollars. So now I have two equations here that I can use in a method called elimination. And to do that, basically, we're just going to try to find a way to manipulate these equations where if we add them up, we will eliminate one of the variables. So to do that, I'll have equation number one, I'll multiply it by positive four. So this will be the entire equation. So I'll have four multiplied by open bracket. Seven X plus six, Y equals 627 and then close the bracket outside of the answer because the answer is also being multiplied. And equation number two will be multiplied by negative three. So we'll have negative three multiplied open bracket, three, X plus eight, Y equals 589. And we close the bracket after the answer. So we can now distribute all that. So we'll have that equation. One will be four multiplied by seven X which is 28 X plus four, multiplied by six Y which is 24 Y that'll be equal to four multiplied by 627 which will be 2508. And I'll place each equation one on top of the other. So I'll have equation number one on top and I'll have equation number two under it. And equation number two will be negative three multiplied by three X which will be negative nine X. And then we'll have negative three multiplied by an eight Y which will be a negative 24. So we'll have a minus 24 Y equals negative three multiplied by 589 which will be negative 1767. Now, the reason why I write each equation on top of the other is because now I can look at this as adding both equations up. So usually when we're adding things, we are taught to put each number one on top of the other, then write the adding symbol and write the line under each number. So by doing that, we can visualize this a little better for when we add each of the equations. So let's add them up. So we'll have 20 X plus negative nine X which is 28 X minus nine X, which will be 19 X. And then we'll have a 24 Y minus 24 Y which will be zero. And that's where the elimination comes in. So we eliminated the variable Y. So we'll have that 19 X equals minus 1767 which is 741. So we'll have 19 X equals 741. And now if we divide both sides by 19, we'll have that X is equal to and this will be $39. And like we said, this is for the price of dog food. So now that we have that X, we can go ahead and continue plugging in. So we'll plug this into equation number two and we'll have that three multiplied by 39 plus eight, Y equals 589. And this was the very first version of equation two. So once we distribute the three and the 39 we'll have 100 and plus eight Y equals 589. And then if we bring the 117 from the left hand side to the right hand side by subtracting, we'll have that A Y is equal to 472. And now we divide both sides by eight to give us that Y is equal to $59. And that will be the price for the cat food, as we mentioned earlier. So we'll have that one sack of dog food is $39.01 sack of cat food is $59 or answer choice B in this question. So I hope that was helpful. And until next time.

Solve each problem. See Examples 5 and 9. At the Berger ranch, 6 goats and 5 sheep sell for \$305, while 2 goats and 9 sheep sell for \$285. Find the cost of a single goat and of a single sheep.

Key Concepts

Systems of Linear Equations

Setting Up Equations from Word Problems

Methods for Solving Systems (Substitution or Elimination)

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