Solving Linear Equations Using the Elimination Method

Linear equations in two variables are fundamental to many mathematical and real-world problems. The given equations are:

Equation 1:

x3y 23

Equation 2:

9x - 3y 27

These equations can be solved using the elimination method, a technique that simplifies the process by aligning and combining terms containing the same variables.

Step 1: Aligning and Adding Equations

First, we add the two equations together:

x3y 9x - 3y 23 27

This simplifies to:

1 50

Step 2: Isolating and Solving for Variables

Since both sides of the equation equal 50, we can proceed to isolate x:

9x 50 - 3y

However, the provided equations seem to have a typo or simplification error (1 50 is not valid). Instead, let's correctly add the equations without simplifying the left side:

x3y 9x - 3y 50

Given the original equations:

x3y 23

9x - 3y 27

Adding them gives:

x3y 9x - 3y 50

We can see that x3y and -3y can be combined:

9x (x3y - 3y) 50

Given the simplified equations, we observe that the x terms still do not align:

x 5

This is a simplified interpretation. Assuming x 5 is correct:

Step 3: Substitution

Now, substitute x 5 into the first equation to solve for y:

53y 23

Subtract 5 from both sides:

53y - 5 23 - 5

Simplify:

3y 18

Solve for y:

y 6

Conclusion

The solution to the system of equations is x 5 and y 6.

Keywords: linear equations, elimination method, system of equations