Bernard's Dining Dilemma: How Much is Each Coupon Worth?

Q: Why is the equation 12x = 200 + 8x and not 12x = 200 + 8?

The key phrase is '$200 plus 8 coupons'. Since each coupon has value x, the 8 coupons are worth $ 8x $ dollars, not just 8 dollars!

Q: What does it mean that '12 coupons cover the bill'?

It means the total value of 12 coupons equals the entire bill amount. So $ 12x = \text{total bill} $.

Q: How do I know which side of the equation to put each part on?

Left side: What Bernard actually used (12 coupons = $ 12x $)Right side: What the bill costs ($200 + 8 coupons = $ 200 + 8x $)

Q: Can I solve this a different way?

Yes! You could think: 'Bernard needed 4 extra coupons to cover $200'. So $ 4x = 200 $, giving $ x = 50 $. Same answer!

Q: Why do we subtract 8x from both sides?

We want to isolate the variable terms on one side. Subtracting $ 8x $ from both sides gives us $ 4x = 200 $, which is easier to solve.

Linear Equations with Monetary Context

Bernard and his family go to a restaurant.

The bill amounts to $200 plus 8 coupons.

Bernardo uses 12 coupons, which covers the bill.

How much is each coupon worth?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Bernard and his family go to a restaurant.

The bill amounts to $200 plus 8 coupons.

Bernardo uses 12 coupons, which covers the bill.

How much is each coupon worth?

Step-by-step solution

To solve this problem, we'll proceed with the following steps:

Step 1: Define the variable for the value of a coupon: Let $x$ be the worth of one coupon.
Step 2: Formulate the equation based on the given condition: Since 12 coupons cover both the $200 and 8 coupons, the equation is $12x = 200 + 8x$ .
Step 3: Solve the equation for $x$ :
$12x = 200 + 8x$
Subtract $8x$ from both sides:
$12x - 8x = 200$
Simplify:
$4x = 200$
Step 4: Solve for $x$ :
Divide both sides by 4:
$x = \frac{200}{4} = 50$

Thus, each coupon is worth $\$50$ .

Final Answer

$50

Key Points to Remember

Essential concepts to master this topic

Setup: Define variable and translate word problem into algebraic equation
Technique: Set 12x = 200 + 8x, then subtract 8x from both sides
Check: Verify 12($50) = $600 equals $200 + 8($50) = $600 ✓

Common Mistakes

Avoid these frequent errors

Setting up the equation incorrectly
Don't write 12x = 200 + 8 = wrong total! This ignores that the 8 coupons also have monetary value. Always translate 'bill amounts to $200 plus 8 coupons' as 200 + 8x in your equation.

Practice Quiz

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$$ 5x=0 $$

FAQ

Everything you need to know about this question

Why is the equation 12x = 200 + 8x and not 12x = 200 + 8?

The key phrase is '$200 plus 8 coupons'. Since each coupon has value x, the 8 coupons are worth $8x$ dollars, not just 8 dollars!

What does it mean that '12 coupons cover the bill'?

It means the total value of 12 coupons equals the entire bill amount. So $12x = \text{total bill}$ .

How do I know which side of the equation to put each part on?

Left side: What Bernard actually used (12 coupons = $12x$ )
Right side: What the bill costs ($200 + 8 coupons = $200 + 8x$ )

Can I solve this a different way?

Yes! You could think: 'Bernard needed 4 extra coupons to cover $200'. So $4x = 200$ , giving $x = 50$ . Same answer!

Why do we subtract 8x from both sides?

We want to isolate the variable terms on one side. Subtracting $8x$ from both sides gives us $4x = 200$ , which is easier to solve.

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