Final answer:
To solve the given system of equations using the elimination method, multiply and combine equations to eliminate one variable, then solve the resulting system of two equations using the elimination or substitution method. The solution is x = 0.55 and y = -7.8875.
Step-by-step explanation:
To solve the given system of equations:
3x - y = 16
x + 4y = -31
-5x - 9y = 23
We can use the method of elimination or substitution to find the values of x and y that satisfy all three equations. Let's use the elimination method:
- Multiply the second equation by 3: 3(x + 4y) = 3(-31) -> 3x + 12y = -93
- Add the first and second equation: (3x - y) + (3x + 12y) = 16 + (-93) -> 6x + 11y = -77
- Multiply the first equation by 5: 5(3x - y) = 5(16) -> 15x - 5y = 80
- Add the third equation and the multiplied first equation: (15x - 5y) + (-5x - 9y) = 80 + 23 -> 10x - 14y = 103
We now have a system of two equations:
6x + 11y = -77
10x - 14y = 103
We can now solve this system using the elimination or substitution method. I will use the elimination method:
- Multiply the first equation by 2: 2(6x + 11y) = 2(-77) -> 12x + 22y = -154
- Add the second equation and the multiplied first equation: (12x + 22y) + (10x - 14y) = -154 + 103 -> 22x + 8y = -51
- Divide the entire equation by 2 to simplify: (22x + 8y)/2 = -51/2 -> 11x + 4y = -25.5
We now have a new equation:
11x + 4y = -25.5
We can solve this equation alongside any of the previous two equations using substitution or elimination to find the values of x and y. I will use substitution by isolating x in the second equation:
x + 4y = -31
- Isolate x: x = -31 - 4y
- Substitute the value of x in the new equation: 11(-31 - 4y) + 4y = -25.5
- Solve for y: -341 - 44y + 4y = -25.5 -> -40y = 315.5 -> y = -7.8875
- Substitute the value of y in the second equation: x + 4(-7.8875) = -31 -> x - 31.55 = -31 -> x = -31 + 31.55 -> x = 0.55
The solution to the system of equations is x = 0.55 and y = -7.8875.