do the equations y=2x+3 and x=1-2y have common solution, no common solution or infinitely many common solutions?
Y = 2x + 3
Eq1: -2x + y = 3
X = 1 - 2y
Eq2: x + 2y = 1
Multiply Eq2 by 2 and add Eq1 and Eq2:
-2x + y = 3
+2x + 4y = 2
sum: 5y = 5
Y = 1
In Eq2, replace Y with 1 and solve for x:
x + 2*1 = 1
X = -1.
Solution set: (x,y) = (-1,1).
One common solution.
Well, let's find out, shall we? It's like a math detective investigation!
First, let's solve y=2x+3:
If we substitute x=1-2y into this equation, we get y = 2(1-2y) + 3, which simplifies to y = 2 - 4y + 3 or 5y = -5, which leads us to y = -1.
Now, let's solve x=1-2y:
If we substitute y=-1 into this equation, we get x = 1 - 2(-1), which simplifies to x = 3.
So, the common solution is (x,y) = (3,-1).
Therefore, the two equations do have a common solution! They make quite a duo; they found each other in the math universe, it seems.
To find out if the equations y=2x+3 and x=1-2y have a common solution, we can substitute the value of one variable from one equation into the other equation.
Let's start by substituting the value of y from the first equation into the second equation:
x = 1 - 2y
x = 1 - 2(2x + 3)
x = 1 - 4x - 6
5x = -5
x = -1
Now, substitute this value of x back into the first equation:
y = 2x + 3
y = 2(-1) + 3
y = -2 + 3
y = 1
So, the values of x and y that satisfy both equations are x = -1 and y = 1.
Therefore, these equations have a common solution.
To determine if the equations y=2x+3 and x=1-2y have a common solution, we can substitute one equation into the other and see if it satisfies both equations.
Let's start by substituting the value of x from the second equation into the first equation:
y = 2(1-2y) + 3
Simplifying the equation, we have:
y = 2 - 4y + 3
Combining like terms, we get:
y = -4y + 5
Now, let's bring all y terms to one side:
y + 4y = 5
Simplifying further, we have:
5y = 5
Dividing both sides by 5, we find:
y = 1
Now, substitute this value of y back into the second equation:
x = 1 - 2(1)
x = 1 - 2
x = -1
So, the solution to the two equations is x = -1 and y = 1.
Therefore, the equations y=2x+3 and x=1-2y have a common solution, which is x = -1 and y = 1.