How to Solve a Linear Equation in Two Variables?

To solve a system of linear equations with two variables, you typically aim to find the values of both variables that satisfy both equations simultaneously. One common method is substitution, which involves the following steps:

Solving Linear Equations by Substitution

The substitution method effectively isolates one variable in terms of the other, allowing you to reduce the problem to a single-variable equation. Here’s a breakdown of the process:

1. Solve for One Variable: Choose one of the two equations and solve it for one variable in terms of the other. This means isolating one variable on one side of the equation. (Reference 1)
2. Substitute: Substitute the expression you found in step 1 into the other equation. This will result in a new equation that contains only one variable. (Reference 2)
3. Solve for the Remaining Variable: Solve the equation you obtained in step 2 for the single variable it contains. This will give you the numerical value of that variable. (Reference 3)
4. Back-Substitute: Substitute the value you found in step 3 back into either of the original equations (or the equation you manipulated in step 1) to solve for the other variable. (Reference 4)

Example

Let's say you have these equations:

• Equation 1: x + y = 5
• Equation 2: 2x - y = 1

Here's how to solve using the steps above:

1. Solve for One Variable: From Equation 1, we can easily solve for x:
   x = 5 - y
2. Substitute: Substitute this expression for x into Equation 2:
   2(5 - y) - y = 1
3. Solve for the Remaining Variable: Simplify and solve for y:
   10 - 2y - y = 1
   10 - 3y = 1
   -3y = -9
   y = 3
4. Back-Substitute: Substitute the value of y (which is 3) back into either Equation 1 or Equation 2 to solve for x. Using Equation 1:
   x + 3 = 5
   x = 2

Therefore, the solution to the system of equations is x = 2 and y = 3.