How to do Linear Equations in Two Variables?

Linear equations in two variables can be solved using various methods. A linear equation in two variables can be written in the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are not equal to zero. Examples include 10x + 4y = 3 and -x + 5y = 2. Solving involves finding the values of x and y that satisfy the equation(s). If you have two linear equations (a system), here's how to approach solving them:

Methods to Solve Systems of Linear Equations

Here's a breakdown of common methods:

• Substitution Method:

  1. Solve one equation for one variable in terms of the other. For example, solve the first equation for x in terms of y (or vice versa).
  2. Substitute the expression obtained in step 1 into the other equation. This will result in a single equation with one variable.
  3. Solve the resulting equation for the remaining variable.
  4. Substitute the value found in step 3 back into either of the original equations (or the expression from step 1) to solve for the other variable.

  Example:

  Solve the system:

  • x + y = 5
  • 2x - y = 1
  1. Solve the first equation for x: x = 5 - y
  2. Substitute into the second equation: 2(5 - y) - y = 1
  3. Simplify and solve for y: 10 - 2y - y = 1 => -3y = -9 => y = 3
  4. Substitute y = 3 back into x = 5 - y: x = 5 - 3 => x = 2

  Solution: x = 2, y = 3

• Elimination Method (also known as the Addition Method):

  1. Multiply one or both equations by a constant so that the coefficients of either x or y are opposites (additive inverses).
  2. Add the two equations together. This eliminates one of the variables.
  3. Solve the resulting equation for the remaining variable.
  4. Substitute the value found in step 3 back into either of the original equations to solve for the other variable.

  Example:

  Solve the system:

  • x + y = 5
  • 2x - y = 1
  1. Notice that the y coefficients are already opposites (+1 and -1).
  2. Add the equations: (x + y) + (2x - y) = 5 + 1 => 3x = 6
  3. Solve for x: x = 2
  4. Substitute x = 2 back into x + y = 5: 2 + y = 5 => y = 3

  Solution: x = 2, y = 3

• Graphing Method:

  1. Rewrite each equation in slope-intercept form (y = mx + b).
  2. Graph each equation on the same coordinate plane.
  3. The point where the two lines intersect is the solution to the system.
  4. If the lines are parallel, there is no solution. If the lines coincide (are the same line), there are infinitely many solutions.

  Note: While visually helpful, this method can be less precise if the intersection point has non-integer coordinates.

When to Use Each Method

• Substitution: Best when one equation is already solved for one variable, or when it's easy to solve for one variable.
• Elimination: Best when the coefficients of one variable are easily made opposites, or when equations are in standard form (ax + by = c).
• Graphing: Useful for visualizing the solution and understanding the relationship between the equations, but less practical for precise solutions.