Solve this system using elimination. If a single solution exists, write the solution as an ordered pair. Your answer will be an ordered pair, no solution, or infinitely many solutions
3a+5b=-19
5a-27b=145
3a+5b=-19. 5a-27b=145
To solve the system using elimination, we will multiply each equation by a suitable number to make the coefficients of one variable opposite in sign. Let's start by eliminating the "a" variable.
First, let's multiply the first equation by 5 and the second equation by 3 to make the coefficients of "a" the same:
5 * (3a + 5b) = 5 * (-19)
3 * (5a - 27b) = 3 * (145)
This simplifies the system to:
15a + 25b = -95
15a - 81b = 435
Now, subtract the second equation from the first equation to eliminate the "a" variable:
(15a + 25b) - (15a - 81b) = -95 - 435
This simplifies to:
15a + 25b - 15a + 81b = -530
Combine like terms:
106b = -530
Divide both sides of the equation by 106:
b = -5
Now that we have found the value of "b", we can substitute it back into one of the original equations to solve for "a". Let's use the first equation:
3a + 5b = -19
3a + 5(-5) = -19
3a - 25 = -19
3a = 6
a = 2
So the solution to the system is (a, b) = (2, -5).
The system has a single solution, and the solution is (2, -5).
To solve this system using elimination, we need to eliminate one variable by manipulating the equations.
To start, let's multiply the first equation by 5 and the second equation by 3 to make the coefficient of 'a' the same in both equations:
5(3a + 5b) = 5(-19) => 15a + 25b = -95 --- Equation (1)
3(5a - 27b) = 3(145) => 15a - 81b = 435 --- Equation (2)
Now, let's eliminate the variable 'a' by subtracting Equation (2) from Equation (1):
(15a + 25b) - (15a - 81b) = -95 - 435
15a + 25b - 15a + 81b = -530
106b = -530
Now we can solve for 'b':
b = -530 / 106
b = -5
After finding the value of 'b', we substitute it back into one of the original equations. Let's use the first equation:
3a + 5(-5) = -19
3a - 25 = -19
3a = 6
a = 2
Therefore, the solution to the system of equations is:
(a, b) = (2, -5)
Hence, the ordered pair solution is (2, -5).