Economics Andre bought 11 books at a used book sale. Some cost 25ϕeach; the others cost 35€ each

step 1 The problem states that there was 11 books bought. So one of my equations is going to be equal t

Economics Andre bought 11 books at a used book sale. Some...

Economics Andre bought 11 books at a used book sale. Some cost 25$\phi$ each; the others cost 35$€$ each. Andre spent $\$ 3.15 .$
a. Write a system of two equations to describe this situation, and solve it by substitution.
b. How many books did Andre buy for each price?

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Key Concepts

System of Linear Equations

A system of linear equations consists of two or more equations that share the same set of unknowns. In problems like this, one equation represents the total number of items purchased while the other represents the total cost, allowing us to determine the quantities of each type of item.

Substitution Method

The substitution method is a procedure for solving systems of equations by isolating one variable in one equation and substituting that expression into the other equation. This method reduces the system to a single equation with one unknown, making it simpler to solve.

Budget Constraint

A budget constraint is an economic concept that represents all possible combinations of goods or services that a consumer can purchase with a given income. In this scenario, it is modeled by the equation that equates the total cost of books to the amount spent.

Translating a Word Problem into Equations

This concept involves reading a real-world problem and identifying the relationships among quantities, such as the number of items and their prices. It includes assigning variables to unknown quantities and setting up equations that accurately describe the constraints and totals provided in the problem.

Transcript

00:02 The problem states that there was 11 books bought.

00:06 So one of my equations is going to be equal to 11.

00:09 Now further reading this problem, the books are at two separate price points.

00:14 So i'm going to represent that as x and y.

00:19 Hence, x plus y equals 11.

00:21 That's going to be my first equation.

00:23 My next equation is going to deal with all of the money.

00:27 So my first type of book, let's just say is x, that's at a price point of 12.

00:32 So i'm going to show that by multiplying the two, plus the other type of book was at 35 cents.

00:42 Again, multiplied by y.

00:44 And it says that he spent a total of $3 .15.

00:51 There's my system of equations.

00:53 Now this says to solve by substitution.

00:56 Well, to solve by substitution, we need one of the equations to say either x equals or y equals.

01:05 I'm going to take this first equation up here and rewrite it so it says x equals and i can do that by thinking i'm sorry by subtracting y from both sides so my new equation is x equals 11 minus y let's substitute in for x all of this will go here let's rewrite this 0 .25 multiplied by the quantity, 11 minus y, plus 0 .35y equals 3 .15.

01:49 Now let's distribute this 0 .25 to both terms.

01:58 So 0 .25 times 11 is 2 .75, 0 .25 times y, plus 0 .25 times y, plus 0 .35 .5 .5 times y, plus 0 .35...

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