what is the solution of the system?
4x-y= -3
6x-y= -8
(5,5.5)
(-2.5, -7)
(-2.5, 2.2)
(20, -28)
what is the solution of the system
5x+6y=20
9x+11y=32
(28, -20)
(28, 20)
(-28, 20)
(20, -28)
4x-y= -3
6x-y= -8
subtract the top from the bottom and you get
2x = -5
x = -5/2
y = 4x+3 = -10+3 = -7
So, (-2.5,-7)
You can do the other in a similar way, or use substitution.
Once you have your answer, check to make sure it works in both equations.
Well, it's certainly not (28, -20) because that would require some creative math skills! And it's definitely not (28, 20) either, because that would make y have a non-integer value. So we can rule out these two options, leaving us with (-28, 20) and (20, -28).
Now, let's check if either of these points satisfy both equations. If we plug in (-28, 20) into the first equation, we get:
5*(-28) + 6*20 = -140 + 120 = -20, which is not equal to 20.
So, (-28, 20) is not a solution to the system of equations.
Now, let's try (20, -28) in the second equation:
9*20 + 11*(-28) = 180 - 308 = -128, which is also not equal to 32.
So, (20, -28) is not a solution either.
Looks like this system doesn't have a solution amongst the given options! But don't worry, solving systems of equations can be a complex task, it's no joke!
To find the solution to a system of equations, we need to determine the values for x and y that satisfy both equations simultaneously. Let's solve each system one by one to find the correct solution.
System 1: 4x - y = -3 and 6x - y = -8
To solve this system, we can use the method of substitution or elimination. Let's use the elimination method:
Step 1: Multiply the first equation by 6 and the second equation by 4 to eliminate the variable y.
6*(4x - y) = 6*(-3)
4*(6x - y) = 4*(-8)
Simplifying these equations gives us:
24x - 6y = -18
24x - 4y = -32
Step 2: Subtract the second equation from the first equation to eliminate x.
(24x - 6y) - (24x - 4y) = -18 - (-32)
-6y + 4y = -18 + 32
-2y = 14
y = -7
Step 3: Substitute the value of y into one of the original equations to find x.
4x - (-7) = -3
4x + 7 = -3
4x = -10
x = -2.5
Therefore, the solution to the system of equations is (-2.5, -7).
System 2: 5x + 6y = 20 and 9x + 11y = 32
Using the same method, let's solve this system:
Step 1: Multiply the first equation by 9 and the second equation by 5 to eliminate the variable x.
9*(5x + 6y) = 9*20
5*(9x + 11y) = 5*32
Simplifying these equations gives us:
45x + 54y = 180
45x + 55y = 160
Step 2: Subtract the second equation from the first equation to eliminate x.
(45x + 54y) - (45x + 55y) = 180 - 160
54y - 55y = 20
-y = 20
y = -20
Step 3: Substitute the value of y into one of the original equations to find x.
5x + 6(-20) = 20
5x - 120 = 20
5x = 140
x = 28
Therefore, the solution to the system of equations is (28, -20).
In conclusion:
- The solution to System 1 is (-2.5, -7).
- The solution to System 2 is (28, -20).
To find the solution of a system of equations, we need to determine the values of x and y that satisfy both equations simultaneously. Let's solve each system step by step:
For the first system:
1. We have two equations:
4x - y = -3 ---(1)
6x - y = -8 ---(2)
2. We can use the method of substitution or elimination to solve the system. Let's use the elimination method, where we eliminate one variable by adding or subtracting the equations.
3. Multiply equation (1) by -1:
-1(4x - y) = -1(-3)
-4x + y = 3 ---(3)
4. Subtract equation (2) from equation (3):
(-4x + y) - (6x - y) = 3 - (-8)
-4x + y - 6x + y = 3 + 8
-10x + 2y = 11 ---(4)
5. We now have a new equation (4) that only involves x and y. Solve it simultaneously with equation (1) or (2) to find x and y.
6. We can either solve by substitution or elimination method again. Let's use the substitution method:
From equation (1), solve for y:
y = 4x + 3
Substitute this value of y in equation (4):
-10x + 2(4x + 3) = 11
-10x + 8x + 6 = 11
-2x = 5
x = -5/2 = -2.5
7. Substitute the value of x back into equation (1) or (2) to find y.
Using equation (1):
4(-2.5) - y = -3
-10 - y = -3
y = -3 + 10
y = 7
Therefore, the solution to the first system is (-2.5, 7).
For the second system:
1. We have the equations:
5x + 6y = 20 ---(5)
9x + 11y = 32 ---(6)
2. Again, we can use the elimination method to solve the system.
3. Multiply equation (5) by 9 and equation (6) by 5:
9(5x + 6y) = 9(20)
5(9x + 11y) = 5(32)
45x + 54y = 180 ---(7)
45x + 55y = 160 ---(8)
4. Subtract equation (7) from equation (8):
(45x + 55y) - (45x + 54y) = 160 - 180
45x + 55y - 45x - 54y = -20
y = -20
5. Substitute the value of y back into equation (5) or (6) to find x.
Using equation (5):
5x + 6(-20) = 20
5x - 120 = 20
5x = 20 + 120
5x = 140
x = 140/5
x = 28
Therefore, the solution to the second system is (28, -20).
In summary, the solutions to the given systems of equations are:
1. (x, y) = (-2.5, 7) for the first system.
2. (x, y) = (28, -20) for the second system.