Solve the equation simultaneously.1/3x+y =3,x+1/2y=4
I want you to solve 1/3x±y=3 x+1/2=4
To solve the given system of equations simultaneously:
Step 1: Begin by multiplying both sides of the first equation by 3 to eliminate the fraction:
(1/3)x + y = 3
3 * [(1/3)x + y] = 3 * 3
x + 3y = 9 (Equation 1)
Step 2: Next, multiply both sides of the second equation by 2 to eliminate the fraction:
x + (1/2)y = 4
2 * [x + (1/2)y] = 2 * 4
2x + y = 8 (Equation 2)
Now, we have a system of two linear equations:
Equation 1: x + 3y = 9
Equation 2: 2x + y = 8
Step 3: Solve the system of equations by using either the substitution or elimination method. In this case, we will use the elimination method:
To eliminate the y-term, we will multiply Equation 1 by -1 and add both equations together:
-1 * (x + 3y) = -1 * 9
-x - 3y = -9
2x + y = 8
Adding both equations:
(-x + 2x) + (-3y + y) = (-9 + 8)
x - 2y = -1 (Equation 3)
Now, we have a new equation:
Equation 3: x - 2y = -1
Step 4: Solve the new equation for x or y. We will solve Equation 3 for x:
x - 2y = -1
x = -1 + 2y (Equation 4)
Step 5: Substitute Equation 4 into one of the original equations (Equation 1 or Equation 2) to solve for y. Let's substitute Equation 4 into Equation 1:
(-1 + 2y) + 3y = 9
-1 + 2y + 3y = 9
5y - 1 = 9
5y = 10
y = 2
Step 6: Substitute the value of y back into Equation 4 to find the value of x:
x = -1 + 2(2)
x = -1 + 4
x = 3
So, the solution to the given system of equations is:
x = 3
y = 2