How do you solve the following system?: #-x -y =2, x -y = -1#

Question

Genesis Allen · Accepted Answer

Let's solve the following equations: #-x-y=2#, #-y=2+x#, #color(red)(y=-2-x#). We already know that #-y#, #x+2+x=-1#, #2x=-1-2#, #2x=-3#, and #x=-3/2#. Color(red)(x=-1.5#, #y=-2-x#, #y=-2-1.5#, and Color(red)(y=-0.5#).

Savannah Anderson · Answer

undefined To solve the system, we can use the method of substitution or elimination. Let's use elimination:

Given the equations:
1. -x - y = 2
2. x - y = -1

Adding equation 1 and equation 2, we get:
(-x - y) + (x - y) = 2 + (-1)
Simplifying:
-x - y + x - y = 1
Combining like terms:
-2y = 1
Divide both sides by -2:
y = -1/2

Now, substitute the value of y into either equation 1 or equation 2. Let's use equation 2:
x - (-1/2) = -1
Simplify:
x + 1/2 = -1
Subtract 1/2 from both sides:
x = -1 - 1/2
x = -3/2

So, the solution to the system is x = -3/2 and y = -1/2.

Eli Assouline · Answer

undefined To solve the system of equations $-x - y = 2$ and $x - y = -1$, you can use the method of substitution or elimination.

Using substitution:
1. Solve one of the equations for one variable. For example, from the second equation, $x = y - 1$.
2. Substitute the expression for the solved variable into the other equation. In this case, substitute $x = y - 1$ into the first equation: $- (y - 1) - y = 2$.
3. Simplify the equation and solve for the remaining variable. $y - 1 - y = 2 \Rightarrow -1 = 2$, which is not true. This indicates that the system has no solution.

Using elimination:
1. Add the two equations together to eliminate one variable. $(-x - y) + (x - y) = 2 + (-1) \Rightarrow -2y = 1 \Rightarrow y = -\frac{1}{2}$.
2. Substitute the value of $y$ back into one of the original equations to solve for $x$. Using the first equation: $-x - (-\frac{1}{2}) = 2 \Rightarrow -x + \frac{1}{2} = 2 \Rightarrow -x = \frac{3}{2} \Rightarrow x = -\frac{3}{2}$.

So, the solution to the system is $x = -\frac{3}{2}$ and $y = -\frac{1}{2}$.