Solve by elimination method.
2x+ 3y = 1
4x + 6y= 2
To solve the given system of equations using the elimination method, we need to eliminate one of the variables by manipulating the equations. Let's start by multiplying the first equation by -2:
-2(2x + 3y) = -2(1)
-4x - 6y = -2
Now, we can rewrite the system of equations with the modified equation:
-4x - 6y = -2
4x + 6y = 2
Next, we can add the equations together to eliminate the variable y:
(-4x - 6y) + (4x + 6y) = -2 + 2
0 = 0
The result we obtained is 0 = 0, which indicates infinitely many solutions. This means that the two equations represent the same line and are linearly dependent. Thus, the system is consistent and dependent.
To solve the given system of equations using the elimination method, you need to eliminate one of the variables by multiplying one or both equations by suitable constant(s) so that the coefficients of one variable in both equations are the same or multiples of each other. This allows you to subtract or add the equations together to eliminate the variable.
Let's solve the given system step by step:
1. Arrange the equations in standard form, if necessary:
2x + 3y = 1 (Equation 1)
4x + 6y = 2 (Equation 2)
2. Notice that the coefficients of x in both equations are already the same, so we can eliminate x by subtracting the two equations.
Multiply Equation 1 by 2 to make the coefficients of y the same:
4x + 6y = 2 (Equation 1 multiplied by 2)
4x + 6y = 2 (Equation 2)
3. Subtract Equation 2 from Equation 1:
(4x + 6y) - (4x + 6y) = 2 - 2
This simplifies to:
0 = 0
4. The equation 0 = 0 indicates that the two original equations are dependent, meaning one equation is a multiple or combination of the other equation. This means that the system has infinitely many solutions.
In conclusion, the given system of equations 2x + 3y = 1 and 4x + 6y = 2 is dependent and has infinitely many solutions.
2x + 3y = 1 times 2 is 4x + 6y = 2
4x + 6y = 2 times 1 is 4x + 6y = 2
These two equations are not independent. They are the same line and no intersection is possible.