1) Find the solution to the following system of two linear equations, if it has a solution.
2x-y=3, -x+y=1
2Find the solution to the following system of two linear equations, if it has a solution.
2x+y=4, x+3/2y=3
a. add the equations: x=4, and put that x into either to solve for y.
b. multipy the first equation by 3/2,
3x+3/2 y=6 Now subtract the second equation from that.
2x=1
x=1/2 and solve for y.
I don't think I follow....
To solve the systems of equations, we can use either substitution or elimination method. Let's use the elimination method for both cases.
1) Solve the system of equations:
2x - y = 3 ...(1)
-x + y = 1 ...(2)
Adding equation (1) and equation (2), we eliminate the variable "y":
(2x - y) + (-x + y) = 3 + 1
Combining like terms:
2x - x - y + y = 3 + 1
Simplifying:
x = 4
Substitute the value of x into equation (2):
(-1)(4) + y = 1
-4 + y = 1
Adding 4 to both sides:
y = 5
Therefore, the solution to the system of equations is x = 4 and y = 5.
2) Solve the system of equations:
2x + y = 4 ...(3)
x + (3/2)y = 3 ...(4)
Multiplying equation (4) by 2 will eliminate the fraction:
2(x) + 2((3/2)y) = 2(3)
2x + 3y = 6 ...(5)
Now, we have the following system of equations:
2x + y = 4 ...(3)
2x + 3y = 6 ...(5)
Subtracting equation (3) from equation (5), we eliminate the variable "x":
(2x + 3y) - (2x + y) = 6 - 4
Combining like terms:
2x - 2x + 3y - y = 2
Simplifying:
2y = 2
Dividing by 2 on both sides:
y = 1
Substitute the value of y into equation (3):
2x + 1 = 4
Subtracting 1 from both sides:
2x = 3
Dividing by 2 on both sides:
x = 3/2 or 1.5
Therefore, the solution to the system of equations is x = 1.5 and y = 1.
To solve a system of linear equations, you can use either the substitution method or the elimination method. I will explain how to solve these two systems using the elimination method.
1) System of equations:
2x - y = 3
-x + y = 1
To apply the elimination method, we need to eliminate one variable by adding or subtracting the equations together. In this case, we can eliminate the y-variable if we add the two equations together:
(2x - y) + (-x + y) = 3 + 1
x = 4
Now that we have found the value of x, we can substitute it back into either of the original equations to find the value of y. Let's substitute it into the second equation:
-x + y = 1
-4 + y = 1
y = 1 + 4
y = 5
Therefore, the solution to the system of equations is x = 4 and y = 5.
2) System of equations:
2x + y = 4
x + (3/2)y = 3
Similarly, we need to eliminate one variable to solve the system. In this case, we can eliminate the x-variable by multiplying the second equation by 2:
2x + y = 4
(2)(x + (3/2)y) = (2)(3)
2x + 3y = 6
Now we can subtract the first equation from the second equation to eliminate the x-variable:
(2x + 3y) - (2x + y) = 6 - 4
2y = 2
y = 1
With the value of y, we can substitute it back into either equation to find the value of x. Let's substitute it into the first equation:
2x + y = 4
2x + 1 = 4
2x = 4 - 1
2x = 3
x = 3/2
Therefore, the solution to the system of equations is x = 3/2 and y = 1.