solve using the elimination method. If the system has no solution or infinite number of solutions a state this
4x-6y=30
3x-5y=24
To solve this system of equations using the elimination method, we need to eliminate one variable by multiplying one or both equations by a constant, and then adding or subtracting the equations to eliminate that variable.
Let's start by eliminating the x term. We can do this by multiplying the second equation by 4 and the first equation by 3.
Multiply the second equation by 4:
4(3x - 5y) = 4(24)
12x - 20y = 96
Multiply the first equation by 3:
3(4x - 6y) = 3(30)
12x - 18y = 90
Now we have:
12x - 18y = 90
12x - 20y = 96
To eliminate the x term, we need to subtract one equation from the other. Let's subtract the first equation from the second equation:
(12x - 20y) - (12x - 18y) = 96 - 90
12x - 20y - 12x + 18y = 6
Simplifying the equation gives us:
-2y = 6
Divide both sides by -2:
y = -3
Now we can substitute the value of y back into one of the original equations to solve for x. Let's use the first equation:
4x - 6y = 30
Substituting y = -3:
4x - 6(-3) = 30
4x + 18 = 30
Subtract 18 from both sides:
4x = 12
Divide by 4:
x = 3
The solution to the system of equations is x = 3 and y = -3.
Therefore, the system has a unique solution.
To solve this system of equations using the elimination method, we can eliminate one of the variables by manipulating one or both equations to have the same coefficient for either the x or y term.
We can start by trying to eliminate the x term. To do so, let's multiply both sides of the second equation by 4 and multiply both sides of the first equation by 3 to make the coefficients of x the same.
Given equations:
4x - 6y = 30 ----(1)
3x - 5y = 24 ----(2)
Multiply equation (1) by 3 and equation (2) by 4:
12x - 18y = 90 ----(3)
12x - 20y = 96 ----(4)
Now we have two equations with the same coefficient for x. We can subtract equation (3) from equation (4) to eliminate x.
(12x - 20y) - (12x - 18y) = 96 - 90
12x - 20y - 12x + 18y = 6
-2y = 6
y = 6 / (-2)
y = -3
Now that we have the value of y, we can substitute it back into one of the original equations to solve for x. Let's use equation (1):
4x - 6(-3) = 30
4x + 18 = 30
4x = 30 - 18
4x = 12
x = 12 / 4
x = 3
Therefore, the solution to the system of equations is x = 3 and y = -3.
Since we were able to find a unique solution for both variables, this system has one solution.