Solve using the elimination method. 2x +3y = 2, 4x + 6y =4.
you should realize that by multiplying the first equation by 2 you get exactly your second equation.
What does your text or your notes say about that situation?
To solve the given system of equations using the elimination method, we need to eliminate one variable by adding or subtracting the equations to get a new equation with only one variable.
Let's start by multiplying the first equation by 2 and the second equation by -1 so that when we add the equations, the x-terms will eliminate:
Equation 1: 2x + 3y = 2
Equation 2: -4x - 6y = -4
Now, add the two equations together:
(2x + 3y) + (-4x - 6y) = 2 + (-4)
Simplifying:
-2x - 3y = -2
Now we have a new equation with only one variable. Let's call this Equation 3.
Next, we need to eliminate the same variable in Equation 3 and one of the original equations. We can choose either Equation 1 or Equation 2. For simplicity, let's choose Equation 1.
Multiply Equation 1 by -3 so that the coefficients of x will cancel out:
-3(2x + 3y) = -3(2)
-6x - 9y = -6
Now, add this new equation obtained from Equation 1 to Equation 3:
(-6x - 9y) + (-2x - 3y) = -6 + (-2)
Simplifying:
-8x - 12y = -8
This is another new equation, let's call it Equation 4.
Now we have two equations, Equation 3: -2x - 3y = -2 and Equation 4: -8x - 12y = -8. We will solve this new system of equations using the elimination method again.
Multiply Equation 3 by 4 so that when we add it to Equation 4, the y-terms will eliminate:
4(-2x - 3y) = 4(-2)
-8x - 12y = -8
Adding Equation 3 and Equation 4:
(-8x - 12y) + (-8x - 12y) = -8 + (-8)
Simplifying:
-16x - 24y = -16
This is our final equation. Notice that there are no variables left, which means we can solve for y:
-16x - 24y = -16
To isolate y, divide both sides by -24:
(-16x - 24y) / -24 = -16 / -24
Simplifying:
2x + 3y = 2/3
Now we have the value of y. To find the value of x, substitute the value of y into one of the original equations. Let's use Equation 1:
2x + 3y = 2
Substituting 2/3 for y:
2x + 3(2/3) = 2
Simplifying:
2x + 2 = 2
Subtracting 2 from both sides:
2x = 0
Dividing both sides by 2:
x = 0
Therefore, the solution to the system of equations is x = 0 and y = 2/3.