Use elimination to solve each system of equations: s-t=4 over s+t=2; -4x+2y=8 over 4x-3y=-10
To solve each system of equations using elimination, we need to eliminate one variable from the system by manipulating one or both of the equations. Here's how we can solve each system separately:
System 1:
s - t = 4 (equation 1)
s + t = 2 (equation 2)
We can add equation 1 to equation 2 to eliminate the variable 't'. By adding both equations, we get:
(s - t) + (s + t) = 4 + 2
2s = 6
Divide both sides of the equation by 2 to solve for 's':
2s/2 = 6/2
s = 3
Now substitute the value of 's' into either equation 1 or equation 2 to solve for 't'. Let's substitute it into equation 1:
3 - t = 4
-t = 4 - 3
-t = 1
Multiply both sides of the equation by -1 to solve for 't':
t = -1
Therefore, the solution to the first system of equations is s = 3 and t = -1.
System 2:
-4x + 2y = 8 (equation 1)
4x - 3y = -10 (equation 2)
We need to eliminate either 'x' or 'y'. To eliminate the 'x' variable, we can multiply equation 1 by 1 and equation 2 by 1:
1 * (-4x + 2y) = 1 * 8
1 * (4x - 3y) = 1 * (-10)
Resulting in:
-4x + 2y = 8 (equation 3)
4x - 3y = -10 (equation 4)
Adding equation 3 to equation 4 eliminates the 'x' variable:
(-4x + 2y) + (4x - 3y) = 8 + (-10)
-y = -2
Multiply both sides of the equation by -1 to solve for 'y':
y = 2
Now substitute the value of 'y' into either equation 3 or equation 4 to solve for 'x'. Let's substitute it into equation 3:
-4x + 2(2) = 8
-4x + 4 = 8
-4x = 8 - 4
-4x = 4
Divide both sides of the equation by -4 to solve for 'x':
x = -1
Therefore, the solution to the second system of equations is x = -1 and y = 2.
In conclusion, the solutions for the systems of equations are as follows:
System 1: s = 3, t = -1
System 2: x = -1, y = 2