3x+6y=21
–5x+2y=13
The given system of linear equations is:
3x + 6y = 21
-5x + 2y = 13
To solve this system, we can use various methods like substitution, elimination, or matrix methods. Let's solve it using the elimination method.
First, we aim to eliminate one variable by making the coefficients of x or y the same (or opposites) in both equations. Let's try to eliminate x by making the coefficients of x in both equations have the same absolute value. We can do this by multiplying the first equation by 5 and the second equation by 3:
(5)(3x + 6y) = (5)(21)
(3)(-5x + 2y) = (3)(13)
This results in:
15x + 30y = 105
-15x + 6y = 39
Now, we can add the two equations to eliminate x:
(15x + 30y) + (-15x + 6y) = 105 + 39
(15x - 15x) + (30y + 6y) = 144
0x + 36y = 144
Now we can simply solve for y:
36y = 144
y = 144 / 36
y = 4
Now that we know y, we can substitute the value of y into one of the original equations to find x. We'll use the first equation:
3x + 6y = 21
3x + 6(4) = 21
3x + 24 = 21
Subtract 24 from both sides:
3x = 21 - 24
3x = -3
Now, divide by 3 to solve for x:
x = -3 / 3
x = -1
So the solution to the system of equations is x = -1, y = 4.