To set up the equation for Lauren's dinner party, we need to represent the total cost of the meals in relation to the number of chicken meals and steak meals she orders. Let \( x \) be the number of chicken dinners and \( y \) be the number of steak dinners.
The cost for chicken dinners is \( 9x \), and the cost for steak dinners is \( 12y \). The total budget is $396.
The equation representing the budget constraint is:
\[ 9x + 12y = 396 \]
Now, we want to rearrange this equation into slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
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Start with the original equation: \[ 9x + 12y = 396 \]
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Isolate the \( y \)-term by subtracting \( 9x \) from both sides: \[ 12y = -9x + 396 \]
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Now, divide every term by 12 to solve for \( y \): \[ y = -\frac{9}{12}x + \frac{396}{12} \]
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Simplify the fractions:
- Simplifying \( -\frac{9}{12} \) gives us \( -\frac{3}{4} \).
- Simplifying \( \frac{396}{12} \) gives us \( 33 \) (since \( 396 \div 12 = 33 \)).
So the equation in slope-intercept form is:
\[ y = -\frac{3}{4}x + 33 \]
This form makes it easier to graph the relationship between chicken and steak dinners Lauren can order, showing how many of each option can fit within her budget.