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Introduction to Systems of Equations - Problem 1
In order to determine if a given point is the solution to a system of equations, plug the x and y values into the two equations. If both equations come out as true -- meaning that when x and y are plugged into the equation, the two sides of the equation equal each other -- then the point is a solution to the system of equations.
A lot of times in Math, you’re given problems that ask you to find an answer that’s going to be a letter or a number. In our case, we have a direction that says determine whether blah, blah, blah. That means my answers are always going to be the words yes or no. Let’s check it out.
Determine whether each point is a solution to this system 2x plus 4y equals 8, 3x plus 4y equals 12. By the way, a lot of the times systems of equations are written with this large curly bracket to show that those 2 equations are grouped together for that system.
Okay, well the way I would know if one of these points or both was a solution is if when I substitute in my x and y values to both equations, both of them come out as true equalities. Here is what I mean. Let’s do part a. My x number is going to be substituted with 2, my y is going to be substituted with 1, so here we go.
2 times my x number plus 4 times my y number, I hope is equal to 8. Let’s see 4 plus 4 equals 8. Okay good, so part a works in the first equation. It also has to work in the second equation so let’s see. 3 times my x number plus 4 times my y number I hope is equal to 12. Is it true that 6 plus 4 equals 12? No. This means that the first point is not a solution to the system. It’s a solution to the top equation, but not to the bottom, therefore it doesn’t count for the whole system no it’s not a solution.
Let’s try the second one. My x numbers are going to be replaced with 4, my Ys are going to be replaced with 0. So I’ll have 2 times 4 plus 4 times y number to see if it’s equal to 8. 8 plus 0 equals 8, all right, it works in my first equation. Now I’m going to try it in my second equation. 3 times my x number plus 4 times my y number should be equal to 12. Is it equal to 12? Yes. Good, this tells me that since (4,0) works in both equations then yes part b that point is a solution to the system of equations.
When you see these kinds of problems, you’re going to plug in your x and y pairs, and make sure it works in both equations that you’re given for your system.