Part 1:
Let's list out what we are given:
x = number of sweaters (each sweater costs $24)
y = number of pants (each pants cost $30)
300 = all the money she will spend
In order to model this situation, we need to make sure that all her spending equals $300. So, how much money will she spend on each type of item? If she buys 10 sweaters, how much would she spend? We have to multiply! 10×$24 = $240 to spend on 10 sweaters. Now, use that same logic to generalize the statement: if she buys x sweaters, she would spend x×$24 = $24x. In the same way, if she buys y pants, she would spend $30y. Now, we can make our equation, since we know that all the money she spends will equal $300:
$24x + $30y = $300
We know that we are dealing with money, so we can drop the $ now:
24x + 30y = 300
Part 2:
Remember that intercepts are where the graph of the equation crosses an axis. So, the x-intercept is where the graph would cross the x-axis. Remember that y=0 for the entire x-axis! So, to solve for the x intercept, we have to plug y=0 into our equation:
24x + 30(0) = 300
24x =300
x = 300/24 = 12.5
Now, to find the meaning of the x-intercept, think of what it means to have y=0. y is the number of pants that she buys. If you plug in y=0, you are saying that she bought 0 pairs of pants. So, the x-intercept tells us how many sweaters she can buy with $300, since she is only buying sweaters! In this case, our answer doesn't make sense (because you can't buy 12 and 1/2 sweaters), but the idea is still the same! If she doesn't buy any pants, she can get 12 sweaters (and have some money left over!)
Part 3:
This is very similar to Part 2. In order to find the y-intercept, you have to plug x=0 into the equation, and then solve for y. And since x=0 means no sweaters, the answer you find will tell you how many pants she can buy if she doesn't get any sweaters! I'll leave the math on this part to you. Feel free to use Part 2 as a guide, and you can comment if you have questions!
Part 4:
Here, we just want a set of sweaters and pants that she could get for $300. To find a set of numbers that make sense, let's solve our equation for y first:
24x + 30y = 300
-24x -24x
30y = 300 - 24x
÷30 ÷30 ÷30
y = 10 - 0.8x
Now, we can test some values of x, and see what we get! We want whole numbers only, since she can't buy pieces of clothes. So, we can try whole numbers for x, starting with x=1:
y = 10 - 0.8(1) = 9.2
x=1 doesn't work, since that gives us y = 9.2, which isn't a whole number. Let's try x=2:
y = 10 - 0.8(2) = 8.4
That didn't work either. But fear not! If you keep trying values of x, you will find two sets of whole numbers! Remember, the value of x is the number of sweaters she can buy, and y is the number of pants. So, once you find those two sets of numbers, you will know two ways that she can spend all of her $300 on a mix of pants and sweaters.
If you have any questions, feel free to comment on this answer!
