Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this.
10x + 7y = -120.5
-20x – 14y = 241
There are three possible cases when we have a system of two equations in two unknowns.
1. There is a unique solution.
2. There is an infinite number of solutions.
3. There is no solution.
Case 1:
Solution can be found by elimination or substitution.
Example:
x+2y=3 ...(1)
x-y =0 ...(2)
Solve by elimination:
(1)-(2)
3y = 3
y=1
Substitute y=1 in (2) to get x=1.
Case 2.
When one equation is a linear combination of the other, we essentially have one equation for two unknowns. There is an infinite set of solutions.
Example:
x+2y = 3
3x+6y = 9
case 3.
When the left-hand-side of one equation is the linear combination of the other, but the right hand side of the equations are not the same, then the equations are not consistent and there is no solution.
Example:
2x+4y = 6
x+2y = 0
See if you can identify to which case the given system belongs.
To solve the system of equations using the elimination method, we need to eliminate one variable by manipulating the two equations. Let's start:
Multiply the first equation by 2 to make the coefficients of x in both equations equal to -20:
2(10x + 7y) = 2(-120.5)
20x + 14y = -241
Now we have:
20x + 14y = -241 (equation 1)
-20x - 14y = 241 (equation 2)
Now, if we add equation 1 and equation 2 together, the x terms will cancel each other out:
(20x + 14y) + (-20x - 14y) = -241 + 241
0 = 0
Since the sum is zero, it means that the two equations are the same line. Therefore, the system has an infinite number of solutions, and every point on the line represents a solution to the system.
In conclusion, the system of equations has an infinite number of solutions.