Solve.
2d−e=7
d+e=5
Use the linear combination method.
The solution is (3, −1) .
The solution is (4, 1).
There are infinitely many solutions.
There is no solution.
adding equations to eliminate e
... 3d = 12
solve for d, then substitute to find e
To solve the system of equations using the linear combination method, we will eliminate one variable. Let's multiply the second equation by 2 so that the coefficients of "e" will cancel out when we add the two equations together.
2d - e = 7
(2)(d + e) = (2)(5)
2d + 2e = 10
Now, let's add the two equations together:
(2d - e) + (2d + 2e) = 7 + 10
4d + e = 17
Now we have a new equation:
4d + e = 17
Let's solve this equation for "e":
e = 17 - 4d
Now substitute this value for "e" into one of the original equations:
d + e = 5
d + (17 - 4d) = 5
d + 17 - 4d = 5
-3d + 17 = 5
-3d = -12
d = 4
Now substitute the value of "d" back into one of the original equations to solve for "e":
2d - e = 7
2(4) - e = 7
8 - e = 7
-e = -1
e = 1
Therefore, the solution to the system of equations is (d, e) = (4, 1).
To solve the given system of equations using the linear combination method, follow these steps:
Step 1: Write down the equations:
2d - e = 7
d + e = 5
Step 2: Multiply one or both equations by a constant, if necessary, to create a situation where adding or subtracting the equations will eliminate one of the variables.
In this case, we can eliminate the variable "e" by adding the two equations together to cancel it out. So, we don't need to multiply any of the equations.
Step 3: Add the equations together:
(2d - e) + (d + e) = 7 + 5
2d - e + d + e = 12
3d = 12
Step 4: Solve for "d" by dividing both sides of the equation by 3:
3d/3 = 12/3
d = 4
Step 5: Substitute the value of "d" back into one of the original equations to solve for the other variable. Let's substitute it into the equation d + e = 5:
4 + e = 5
e = 5 - 4
e = 1
Step 6: Write the solution as an ordered pair (d, e):
The solution is (4, 1).
Therefore, the correct answer is:
The solution is (4, 1).