Solve for Pen Price: 4 Pens + 9 Notebooks = $51 with Double Price Relationship

Q: Why do I use x for the notebook instead of the pen?

Since the pen costs twice as much as the notebook, it's easier to let the notebook = x (the simpler price). Then pen = 2x automatically gives us the relationship without fractions!

Q: How do I know which item to call x?

Always let x represent the cheaper item when one costs a multiple of the other. This avoids fractions like x/2 and keeps your math cleaner.

Q: Can I solve this without substitution?

Yes! You could use two variables (p for pen, n for notebook) and create two equations: 4p + 9n = 51 and p = 2n. But substitution with one variable is usually faster.

Q: How do I check if my final answer makes sense?

Ask yourself: Does the pen cost exactly twice the notebook? In this case, $6 is exactly twice $3, and 4($6) + 9($3) = $51. Both conditions check out!

System of Equations with Variable Substitution

Daniela goes to the bookshop and buys 4 pens and 9 notebooks for a total of $51.

The price of a pen is twice as much as the price of a notebook.

How much is a pen?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Daniela goes to the bookshop and buys 4 pens and 9 notebooks for a total of $51.

The price of a pen is twice as much as the price of a notebook.

How much is a pen?

Step-by-step solution

We will identify the price of the notebook with x and since the price of the pen is 2 times greater we will mark the price of the pen with 2x

The resulting equation is 4 times the price of a pen plus 9 times the price of a notebook = 51

Now we replace and obtain the following equation:

$\( 4\times2x+9\times x=51$

According to the rules of the order of arithmetic operations, multiplication and division operations precede addition and subtraction, therefore we will first solve the two multiplication exercises and then add them up:

$(4\times2x)+(9\times x)=51$

$(4\times2x)=8x$

$(9\times x)=9x$

$8x+9x=17x$

Now the obtained equation is: $17x=51$

We divide both sides by 17 and find x

$x=\frac{51}{17}=3$

As we discovered that x is equal to 3, we will place it accordingly and find out the price of a pen: $2\times x=2\times3=6$

Final Answer

$6$

Key Points to Remember

Essential concepts to master this topic

Variable Definition: Let notebook price = x, then pen price = 2x
Equation Setup: 4 pens + 9 notebooks = 4(2x) + 9x = 51
Verification: Check 4( $6) + 9($ 3) = $24 +$ 27 = $51 ✓

Common Mistakes

Avoid these frequent errors

Setting up incorrect variable relationships
Don't make pen = x and notebook = 2x when pen costs twice as much = backwards pricing! This gives pen = $3 and notebook =$ 6, which contradicts the problem. Always read carefully: if pen costs twice notebook price, then pen = 2x and notebook = x.

Practice Quiz

Test your knowledge with interactive questions

$$ x+7=14 $$

$x=\text{?}$

FAQ

Everything you need to know about this question

Why do I use x for the notebook instead of the pen?

Since the pen costs twice as much as the notebook, it's easier to let the notebook = x (the simpler price). Then pen = 2x automatically gives us the relationship without fractions!

How do I know which item to call x?

Always let x represent the cheaper item when one costs a multiple of the other. This avoids fractions like x/2 and keeps your math cleaner.

What if I set up the equation backwards?

You'll get the wrong answer! If you accidentally make notebook = 2x and pen = x, you'll solve correctly but your final answers will be swapped. Always double-check your variable definitions.

Can I solve this without substitution?

Yes! You could use two variables (p for pen, n for notebook) and create two equations: 4p + 9n = 51 and p = 2n. But substitution with one variable is usually faster.

How do I check if my final answer makes sense?

Ask yourself: Does the pen cost exactly twice the notebook? In this case, $6 is exactly twice$ 3, and 4( $6) + 9($ 3) = $51. Both conditions check out!

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