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lesson 8.4 closure
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1. write a system of equation word problem using yourself as the person in the problem. relate it to something you enjoy doing. remember that you must have 2 unknowns and 2 equations within your word problem. the last sentence should be the question or \what youre looking for\.
2. write the equations that youd use to solve your system.
3. write the solution to the system (use desmos to find this).

Explanation:

1. Word Problem (Relating to Reading and Buying Books, a Hobby)

I enjoy reading novels and comic books. Last week, I bought some novels and comic books. The total number of books I bought is 10. Each novel costs $15 and each comic book costs $10, and I spent a total of $125. How many novels (\(x\)) and how many comic books (\(y\)) did I buy?

2. Equations for the System

Step 1: Define Variables

Let \(x\) be the number of novels and \(y\) be the number of comic books.

Step 2: First Equation (Total Number of Books)

The total number of books is the sum of novels and comic books, which is 10. So, the first equation is:
\[x + y = 10\]

Step 3: Second Equation (Total Cost)

Each novel costs $15, so the cost of LXI0 novels is LXI1. Each comic book costs $10, so the cost of \(y\) comic books is \(10y\). The total cost is $125, so the second equation is:
\[15x + 10y = 125\]

3. Solution to the System (Using Desmos or Substitution/Elimination)

Method: Substitution

From the first equation \(x + y = 10\), we can express \(y\) in terms of \(x\):
\[y = 10 - x\]

Substitute \(y = 10 - x\) into the second equation \(15x + 10y = 125\):
\[15x + 10(10 - x) = 125\]

Step 1: Expand the Equation

\[15x + 100 - 10x = 125\]

Step 2: Combine Like Terms

\[5x + 100 = 125\]

Step 3: Solve for \(x\)

Subtract 100 from both sides:
\[5x = 125 - 100\]
\[5x = 25\]
Divide both sides by 5:
\[x = 5\]

Step 4: Solve for \(y\)

Substitute \(x = 5\) into \(y = 10 - x\):
\[y = 10 - 5\]
\[y = 5\]

Final Answers

1. Word Problem: As constructed above (about buying novels and comic books).
2. Equations: \(x + y = 10\) and \(15x + 10y = 125\) (where \(x\) = number of novels, \(y\) = number of comic books).
3. Solution: \(x = 5\) (5 novels) and \(y = 5\) (5 comic books).

Answer:

1. Word Problem (Relating to Reading and Buying Books, a Hobby)

I enjoy reading novels and comic books. Last week, I bought some novels and comic books. The total number of books I bought is 10. Each novel costs $15 and each comic book costs $10, and I spent a total of $125. How many novels (\(x\)) and how many comic books (\(y\)) did I buy?

2. Equations for the System

Step 1: Define Variables

Let \(x\) be the number of novels and \(y\) be the number of comic books.

Step 2: First Equation (Total Number of Books)

The total number of books is the sum of novels and comic books, which is 10. So, the first equation is:
\[x + y = 10\]

Step 3: Second Equation (Total Cost)

Each novel costs $15, so the cost of LXI0 novels is LXI1. Each comic book costs $10, so the cost of \(y\) comic books is \(10y\). The total cost is $125, so the second equation is:
\[15x + 10y = 125\]

3. Solution to the System (Using Desmos or Substitution/Elimination)

Method: Substitution

From the first equation \(x + y = 10\), we can express \(y\) in terms of \(x\):
\[y = 10 - x\]

Substitute \(y = 10 - x\) into the second equation \(15x + 10y = 125\):
\[15x + 10(10 - x) = 125\]

Step 1: Expand the Equation

\[15x + 100 - 10x = 125\]

Step 2: Combine Like Terms

\[5x + 100 = 125\]

Step 3: Solve for \(x\)

Subtract 100 from both sides:
\[5x = 125 - 100\]
\[5x = 25\]
Divide both sides by 5:
\[x = 5\]

Step 4: Solve for \(y\)

Substitute \(x = 5\) into \(y = 10 - x\):
\[y = 10 - 5\]
\[y = 5\]

Final Answers

1. Word Problem: As constructed above (about buying novels and comic books).
2. Equations: \(x + y = 10\) and \(15x + 10y = 125\) (where \(x\) = number of novels, \(y\) = number of comic books).
3. Solution: \(x = 5\) (5 novels) and \(y = 5\) (5 comic books).