Miguel received a gift card of $$ \$ 100 $$ for a pizza restaurant. The restaurant charges $$ \$ 15 $$ per pizza. Jane received a gift card of $$ \$ 70 $$ for a different pizza restaurant. The restaurant charges $$ \$ 10 $$ per pizza. Let $$ x $$ be the number of pizzas purchased. (a) For each card, write an expression for the amount of money left on the card after purchasing $$ x $$ pizzas. Amount of money left on Miguel's card (in dollars) $$ =\square $$ Amount of money left on Jane's card (in dollars) $$ =\square $$ (b) Write an equation to show when the two cards would have the

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Given:
- Miguel's gift card is worth $100.
- The restaurant charges $15 per pizza.
- Jane's gift card is worth $70.
- The other restaurant charges $10 per pizza.

Let $$ x $$ be the number of pizzas purchased.

(a) To find the amount of money left on each card after purchasing $$ x $$ pizzas, we need to subtract the total cost of pizzas from the initial amount on the gift card.

For Miguel's card:
Amount of money left on Miguel's card = Initial amount on Miguel's card - Total cost of pizzas
Amount of money left on Miguel's card = $100 - $15x

For Jane's card:
Amount of money left on Jane's card = Initial amount on Jane's card - Total cost of pizzas
Amount of money left on Jane's card = $70 - $10x

(b) To find when the two cards would have the same amount of money left, we need to set the expressions for the amount of money left on each card equal to each other and solve for $$ x $$.

Setting the expressions equal to each other:
$100 - 15x = 70 - 10x$

Now, we can solve this equation to find the value of $$ x $$ when the two cards would have the same amount of money left.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$100-15x=70-10x$$
- step1: Move the expression to the left side:
  $$-15x+10x=70-100$$
- step2: Add and subtract:
  $$-5x=70-100$$
- step3: Add and subtract:
  $$-5x=-30$$
- step4: Change the signs:
  $$5x=30$$
- step5: Divide both sides:
  $$\frac{5x}{5}=\frac{30}{5}$$
- step6: Divide the numbers:
  $$x=6$$
The value of $$ x $$ when the two cards would have the same amount of money left is $$ x = 6 $$.
(a) Amount of money left on Miguel's card = $$ 100 - 15x $$ Amount of money left on Jane's card = $$ 70 - 10x $$ (b) When both cards have the same amount left, $$ 100 - 15x = 70 - 10x $$, which occurs when $$ x = 6 $$.

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