Solve the following system using elimination. If a single solution exists, write the solution as an ordered pair. Your answer will be an ordered pair, no solution, or infinitely many solutions
x-y=4
3x-5=2
bring y to the other side on first equation so that you will have
x= y + 4 and substitute y + 4 for x in second equation
3(y+4) - 5 = 2 and solve
3y + 12 - 5 = 2
3y +7 = 2
3y= -5
y= -5/3
plug the answer for y into either equation and solve
To solve the system using elimination, we need to eliminate one of the variables. We can do this by multiplying one or both of the equations by a constant so that the coefficients of one of the variables are the same in both equations.
Let's start by multiplying the first equation by 3, and the second equation by 2 to make the coefficients of x the same:
3(x - y) = 3(4)
2(3x - 5) = 2(2)
This simplifies the system to:
3x - 3y = 12
6x - 10 = 4
Next, we can eliminate y by adding the two equations:
(3x - 3y) + (6x - 10) = 12 + 4
3x - 3y + 6x - 10 = 16
9x - 3y - 10 = 16
Now we can simplify further:
9x - 3y - 10 = 16
9x - 3y = 26
To eliminate the coefficient of y, we can multiply both sides of the second equation by -1/3:
-(9x - 3y) = -26
-9x + 3y = -26
Now, let's add the two equations:
(9x - 3y) + (-9x + 3y) = 26 + (-26)
9x - 9x - 3y + 3y = 0
0 = 0
Since the variables cancel out and we are left with 0 = 0, this implies that the two equations are actually the same equation. Therefore, there are infinitely many solutions to this system.
The solution can be any ordered pair (x, y) that satisfies the original equations.
To solve the given system using elimination, we will eliminate one of the variables by performing operations on the equations. Let's solve it step by step:
1) Rewrite the system of equations:
x - y = 4 (Equation 1)
3x - 5 = 2 (Equation 2)
2) Multiply Equation 1 by 3, so the coefficients of x in both equations will become the same:
3(x - y) = 3(4)
3x - 3y = 12 (New Equation 1)
3) Now we can eliminate the x terms by subtracting the new Equation 1 from Equation 2:
(3x - 5) - (3x - 3y) = 2 - 12
3x - 3x - 5 + 3y = -10
0 + 3y - 3x = -10
3y - 3x = -10 (New Equation 2)
4) Simplify the equation:
3y - 3x = -10
5) Divide both sides of the equation by 3 to simplify it further:
(3y - 3x)/3 = (-10) / 3
y - x = -10/3
y = -10/3 + x
y = -(10/3) + x (New Equation 3)
6) Now we have two equations:
y = -(10/3) + x (New Equation 3)
x - y = 4 (Equation 1)
7) Substitute the value of y from the New Equation 3 into Equation 1:
x - (-(10/3) + x) = 4
x + 10/3 - x = 4
(3x + 10)/3 = 4
3x + 10 = 12
8) Solve for x:
3x = 12 - 10
3x = 2
x = 2/3
9) Substitute the value of x back into Equation 1 to find y:
(2/3) - y = 4
- y = 4 - (2/3)
- y = 12/3 - 2/3
- y = 10/3
y = -10/3
10) So the solution to the system of equations is x = 2/3 and y = -10/3.
The ordered pair representing the solution is (x, y) = (2/3, -10/3).