Solving for x and y in a System of Linear Equations
In this article, we will solve for x and y in the following system of linear equations:
Equation 1: x + y - 2 = 1 Equation 2: 3x - y + 1 = 8
To solve for x and y, we can use the method of substitution or elimination. Here, we will use the elimination method.
Step 1: Multiply Equation 1 by 3 and Equation 2 by 1
To eliminate the y-term in both equations, we will multiply Equation 1 by 3 and Equation 2 by 1:
Equation 1: 3x + 3y - 6 = 3 Equation 2: 3x - y + 1 = 8
Step 2: Add both equations to eliminate the y-term
Now, we will add both equations to eliminate the y-term:
(3x + 3y - 6) + (3x - y + 1) = 3 + 8 Combine like terms: 6x - 3y - 5 = 11
Step 3: Solve for x
Now, we can solve for x by rearranging the equation:
6x - 5 = 11 + 3y 6x = 16 + 3y x = (16 + 3y) / 6
Step 4: Substitute x into one of the original equations to solve for y
Now that we have an expression for x, we can substitute it into one of the original equations to solve for y. We will use Equation 1:
x + y - 2 = 1 ((16 + 3y) / 6) + y - 2 = 1
Step 5: Solve for y
Now, we can solve for y:
((16 + 3y) / 6) + y - 2 = 1 16 + 3y + 6y - 12 = 6 9y = 2 y = 2/9
Step 6: Substitute y back into the expression for x to find the value of x
Now that we have the value of y, we can substitute it back into the expression for x:
x = (16 + 3y) / 6 x = (16 + 3(2/9)) / 6 x = (16 + 6/9) / 6 x = 16/6 + 1/18 x = 8/3 + 1/18 x = (24 + 1) / 18 x = 25/18
Therefore, the values of x and y are:
x = 25/18 y = 2/9
We have successfully solved for x and y in the given system of linear equations.
