## Homework 2 Solving Systems By Substitution Worksheets

Mary Anderson might seem like your ordinary neighbor, except for one thing. She has a mind-bending number of cats. Don't ask us why, or how - she just does.

There's a cat show coming up and the cash rewards are huge. So Mary plans to bring as many cats as possible.

She knows her short-haired cats are better looking than the long-haired ones, so she decides to bring 7 times more short-haired cats than long-haired ones.

Now here's the thing: Mary can only spend a certain amount on cat grooming, and the local stylist isn't cheap. He charges 18 dollars for cats with short hair and 36 dollars for cats with long hair.

Her cat styling budget is $1,944. She needs to figure out how many long- and short-haired cats to bring.

### Setting up a System of Equations

Let's help her by solving **sytems of equations**. Let the variable x denote the number of short-haired cats she brings, and let y be the number of long-hair cats. We'll write two equations to help her.

Grooming short-haired cats cost $18, so 18x represents the cost to groom all the short-haired cats she wants to bring. To groom long-haired cats costs $36, so we can write this as 36y.

Added together, these terms give the total cost of grooming cats for the show. She has a budget of $1,944.

The second equation comes from the fact that she wants to bring 7 times more short-haired cats than long-haired ones.

That means that x, the number of short-haired cats, equals 7 times the number of long-haired cats.

### Solving a System of Equations

Now take a close look at this second equation: it tells us that x has exactly the same value as 7y.

This means we can **substitute** 7y for x in the first equation. Why would we want to do that? Well, now that we only have one variable, we can **solve for y**. We've got two **like terms** on the left, so let's combine them.

- First, 18 times 7y is 126y.
- 126y plus 36y is 162y, so we have 162y = 1944.
- Dividing both sides by 162, we see that y = 12.
- To find x, substitute 12 into our second equation: x = 7y.
- We see x equals 7 times 12, or 84.

Therefore, Mary can bring 12 long-haired cats and 84 short-haired cats. Let's check our work.

### Checking the Solution

If these numbers are correct, then they must satisfy both of the equations. That means if we substitute 12 and 84 for x and y, respectively, then simplify, we should get the same values on either side of the equal signs.

So let's check. After multipying, we have 1512 + 432, giving us 1,944. That's equal to the right-hand side, so these values of x and y satisfy the equation.

In the second equation, we know 84 = 7 * 12, so we're good here, too, which means all our work checks out. Mary can definitely bring 12 long-haired cats and 84 short-haired ones.

Alltogether, that's 96 cats to groom. She brings them to the groomers for same-day service. He basically just blow dries them all at once. She picks them up and finally arrives at the show.

Poor Mary: She missed one important detail: this cat show is only for Sphynx cats - cats without hair. What's this? Mary seems to have an idea...

Один из прямоугольников вдруг закрыла чья-то тень. Даже не взглянув на верхушку башни, Халохот бросился к лестнице. ГЛАВА 99 Фонтейн время от времени стучал кулаком по ладони другой руки, мерил шагами комнату для заседаний, то и дело посматривая на вращающиеся огни шифровалки.

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