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### Course: 8th grade (Illustrative Mathematics) > Unit 3

Lesson 9: Lesson 12: Solutions to linear equations# Solutions to 2-variable equations

The solution of an equation with one variable is a number. How does the solution of a 2-variable equation look like? Well, it is an

*. Learn more about it and how to test solutions to 2-variable equations.***ordered pair**## Want to join the conversation?

- serious question . Does anyone actually use this ? if so for what ? cus honestly i dont see the point(32 votes)
- Yahy,

Solving linear equations is a foundation step for solving systems of linear equations, which is a foundation step for linear programming (which, surprisingly, is not computer programming) or linear optimization.

Linear programming is a method for calculating an optimal result given a set of constraints. Linear programming is used in business all the time.

For example, if you owned a shop, you might want to know the best combination of items to stock in the limited capacity of the shop in order to maximize cash flow. Your most profitable item might not be the best selling or your space might only hold 10 of your most profitable item or 40 of a less profitable item. Linear programming helps you to determine whether to stock 5 of the most profitable item and 20 of the less profitable item or 4 of the most profitable and 30 of the less profitable to keep the cash flowing steadily.

If you don't own the shop, but you work there, you want your boss to be able to do linear programming so there is money for you to be paid.

If you are able to do the linear programming, you would be able to see whether the boss has the right idea and, perhaps, you could make a suggestion that will get you a raise.(74 votes)

- this video helps but I could use more help(11 votes)
- Linear equations are a foundation of learning linear systems. If you can review some previous work you should recognize the relationship between x and y coordinates to grind then on a 2 dimensional coordinate plane. If you need help understanding draw yourself a table labeling x and y axis(3 votes)

- I need some examples of linear equations in 2 variables(9 votes)
- The section at this link deals with that topic: https://www.khanacademy.org/math/algebra/two-var-linear-equations(10 votes)

- I feel like this is pretty simple math but what is it used for in real life?(9 votes)
- Linear equations are used in real life for modeling relationships between variables in fields such as economics, business, engineering, physics, finance, computer science, environmental science, transportation, social sciences, medicine, and sports analytics. They help optimize systems, analyze data, and solve problems across various domains.(5 votes)

- WHY is this so confusing?
**Sinks into Confusion**(8 votes)- Think of it thid way y=mx+b(1 vote)

- serious question . Does anyone actually use this ? if so for what ? cus honestly i dont see the poin(2 votes)
- Sometimes no you won't use a few things you learn in school for in real life. But the whole point of this is so when you do need it, you will notice that you learned it in school!

And even if you can't think of a reason, you will still need to know this to continue in school and learn things that you will need to know for the real world.

And if**that**can't even be used in real life, then just have fun learning!(12 votes)

- I have a question

9b-5=2

what is b?(2 votes)- For the equation, what you do to one side you do to the other. And you want b to "be" alone in order to find out what number it equals!

As such:

9b - 5 = 2 get rid of the 5 by adding 5

+5 +5 remember to add to both sides!

Then you have 9b = 7

9b is actually 9 x b, and in order to remove multiplication you need to divide. So divide both sides by 9.

9b = 7 7 divided by 9 (7/9) is 0.777777... So for simplicity I would leave it as a fraction, unless stated otherwise.

Therefor, you get b = 7/9

Tada!

For a more literal answer, the b in 9b - 5 = 2 is a letter, otherwise known as a**variable**.

Hope this helps!(8 votes)

- How can I solve an equation with two variables? An equation like ' 2x + 4y =100 solve for y and x '(2 votes)
- Solving for a variable is also called isolating the variable, so you want to move everything away from it.

To isolate y, first subtract 2x from both sides (4y = -2x + 100), then divide by 4 (y = -1/2 x + 25). This is the more common usage because this is a linear function in slope intercept form - y in terms of x or y dependent on x.

To solve for x, subtract 4y from both sides (2x = - 4y + 100), then divide by 2 (x = - 2y + 50).(7 votes)

- Is there a way to find two variables without options given

eg:- 4y+5x=12 but without options given(3 votes)- If the goal is to find x and y
**intersection**, one question can't answer that. You must have at least 2 equations to solve the two variables, since you can't have an intersection point w one equation.

hope that helps !(3 votes)

- For example, if you owned a shop, you might want to know the best combination of items to stock in the limited capacity of the shop in order to maximize cash flow. Your most profitable item might not be the best selling or your space might only hold 10 of your most profitable item or 40 of a less profitable item. Linear programming helps you to determine whether to stock 5 of the most profitable item and 20 of the less profitable item or 4 of the most profitable and 30 of the less profitable to keep the cash flowing steadily.(4 votes)
- Yes, you're absolutely right! Linear programming is a mathematical technique used to find the best possible outcome in a given situation with constraints. In the context of your shop example, linear programming can help you determine the optimal combination of items to stock in your limited shop space in order to maximize cash flow.(1 vote)

## Video transcript

- [Voiceover] Which of the ordered pairs is a solution of the following equation? Negative three x minus y is equal to six. What we have to remind ourselves is when we're give an ordered pair, the first number is the x
coordinate and the second number is the y coordinate, or the y value. So when they tell us the ordered pair, negative four comma four, they're saying "hey look, if x is equal to "negative four, and y is
equal to positive four, "does that satisfy this equation?" And what we can do, is
we can just try that out. So we have negative three
and everywhere we see an x, everywhere we see an x, we can replace it with negative four. So it's negative three
times negative four, minus, minus and everywhere we see a y, we can replace it with positive four. We replace it with positive four. So negative three times x minus y, which is four, needs to be equal to six. Needs to be equal to six. Now is this indeed the case? Negative three times negative four is positive 12. Positive 12 minus four,
positive 12 minus four is equal to eight, it's not equal to six. Is not equal, is not equal to six. So this one does not work out. So let's see, negative three comma three. We can do the same thing here. Let's see what happens when
x is equal to negative three and y is equal to positive three. So we substitute back in,
we get negative three. Negative three times x,
which now we're going to try out x being equal to negative three. Minus y, minus y. Y is positive three here. Minus y, gonna do that y color blue. Minus y now needs to be equal to, now needs to be equal, just like before needs to be equal to six. So negative three times negative three. That's going to be positive nine. Nine minus three is indeed equal to six. Nine minus three is
indeed equal to six. Nine minus three is six. That is equal to six. This works out. So negative three comma
three is an ordered pair that is a solution to this equation.