Solve the following system of linear equations by addition. Indicate whether the given system of linear equations has a solution, has no solution or has an infinite number of solutions. Find the solution. -10x+2y=20 -5x+y=25.
To solve the system of linear equations by addition, we will add the two equations together in such a way that one of the variables will cancel out.
We can achieve this by multiplying the second equation by 2, so that the coefficients of y in both equations will have opposite signs:
-10x + 2y = 20
-10x + 2y = 50
Now, let's add the two equations together:
(-10x + 2y) + (-10x + 2y) = 20 + 50
Simplifying the equation:
-20x + 4y = 70
Now, let's solve for y:
4y = 70 + 20x
y = (70 + 20x)/4
y = 17.5 + 5x
Next, let's substitute this value of y into one of the original equations to solve for x. Let's use the first equation:
-10x + 2(17.5 + 5x) = 20
-10x + 35 + 10x = 20
35 = 20
Since this equation is not true, we have arrived at a contradictory statement. Therefore, the given system of linear equations has no solution.