Solve each system of linear equation and explain any method you used:
5x+4y=-1
-7x-2y=-13
To solve this system of linear equations, we can use the method of substitution or the method of elimination.
Method of substitution:
From the first equation, we can solve for x in terms of y:
5x + 4y = -1
5x = -4y - 1
x = (-4y - 1)/5
Now, we substitute this expression for x in the second equation:
-7((-4y - 1)/5) - 2y = -13
(-28y - 7)/5 - 2y = -13
(-28y - 7 - 10y)/5 = -13
(-38y - 7)/5 = -13
To isolate y, we cross-multiply:
-38y - 7 = -65
-38y = -65 + 7
-38y = -58
y = -58/-38
y = 29/19
Now that we have the value of y, we substitute it back into the expression for x:
x = (-4(29/19) - 1)/5
x = (-116/19 - 1)/5
x = (-116/19 - 19/19)/5
x = (-135/19)/5
x = -135/95
x = -27/19
Therefore, the solution to the system of linear equations is x = -27/19 and y = 29/19.
Method of elimination:
To use the method of elimination, we can multiply the first equation by 2 and the second equation by 4 to make the coefficients of x in both equations the same:
10x + 8y = -2
-28x - 8y = -52
Now, we add the two equations together to eliminate y:
(10x + 8y) + (-28x - 8y) = -2 + (-52)
-18x = -54
x = -54/-18
x = 3
Substitute this value of x back into one of the original equations, for example, the first equation:
5(3) + 4y = -1
15 + 4y = -1
4y = -1 - 15
4y = -16
y = -16/4
y = -4
Therefore, the solution to the system of linear equations is x = 3 and y = -4.