Respuesta :
Answer:
(3, -5) or x=3, y=-5
Step-by-step explanation:
Systems of Equations:
In a systems of equation, we're looking for what pair of (x, y) values make both equations true. When a pair of (x, y) makes an equation true, that just means it's on the line of the graph. So a pair of (x, y) values that makes both equations true, is simply when both lines intersect.
Solving Systems of Equations:
There are multiple methods to solve a systems of equation including: elimination, substitution, and graphing.
Each way has their advantages, and I'll show to to apply substitution and elimination
Elimination Method:
In this method we want to manipulate one of equations (or possibly both) so that they each have a variable with the "same coefficient" but different sign (one positive, one negative).
In the equations given, you may notice that the "y" variables each have the same coefficient, so we could multiply either of the equations by negative one, so one has positive y and another has negative y.
The reason we want the equations set up in this way, is so that we can add the equations which results in one of the variables being "eliminated" so that we can solve for a single value for the other variable.
So we start with:
[tex]x+y=-2\\2x+y=1[/tex]
let's multiply the top equation by -1 (remember to apply this to both sides so you don't actually change the equation)
[tex]-(x+y)=-(-2)\implies -x-y=2[/tex]
So now from here we have the two equations
[tex]-x-y=2\\2x+y=1[/tex]
If we add the two equations we get the following
[tex](-x+2x)+(-y+y)=(2+1)[/tex]
Notice how the -y and y cancel each other out? This is the entire point of the elimination method, so now we can solve for x
[tex]x=3[/tex]
This actually simplifies in our favor since -x + 2x = x, and the right side is just some constant, so we don't need to manipulate the equation anymore to solve for x.
Now that we know what the x-value is, we can plug this into either equation to solve for y. For simplicity I'll use the x + y = -2 equation.
[tex]3+y=-2\\y=-2-3\\y=-5[/tex]
So we know the solution is (3, -5) or [tex]x=3,\ y=-5[/tex]
Substitution Method
The substitution method has the same goal as the elimination method, that is, get a one-variable equation so we can solve for one value. In this method, we want to define one variable in terms of the other variable, that way we can substitute this into the other equation, giving us a new equation with only one variable.
So let's just start with the top equation:
[tex]x+y=-2[/tex]
From here we can define "y" in terms of x, by subtracting x from both sides.
[tex]y=-2-x[/tex]
Now let's look at our other equation
[tex]2x+y=1[/tex]
Notice how it has a y variable, well we have one way of defining it: y = -2 - x. If we plug this into the bottom equation, now we only have x terms, giving us a one-variable equation.
[tex]2x+(-2-x)=1[/tex]
Now let's add 2 to both sides to cancel out the -2, and simplify 2x - x
[tex]x=3[/tex]
From here we essentially do the same thinkg we did in elimination method, except it may be a bit easier since we already have "y" solved for, we just need an x-value, which we now have!
[tex]y=-2-x\\\text{we know that x = 3}\\y=-2-(3)\\y=-5[/tex]
so now we have our solution: [tex](3, -5)\text{ or }x=-3, y=-5[/tex]
