How to Solve Linear Equations by Substitution Step by Step?

Solving linear equations by substitution is a straightforward method that involves expressing one variable in terms of another and then substituting it into the other equation. Here's a step-by-step guide:

Step 1: Solve for One Variable

• Choose one of the equations and solve for one variable in terms of the other.
• For example, if you have the equation: 2x + y = 5, you can solve for y:
  • y = 5 - 2x

Step 2: Substitute

• Substitute the expression you found in Step 1 into the other equation.
• In the example above, if the second equation is x - 3y = 1, substitute y = 5 - 2x into it:
  • x - 3(5 - 2x) = 1

Step 3: Solve for the Remaining Variable

• Simplify the equation and solve for the remaining variable.
• In our example, simplify the equation:
  • x - 15 + 6x = 1
  • 7x = 16
  • x = 16/7

Step 4: Substitute to Find the Other Variable

• Substitute the value you found in Step 3 back into either of the original equations to solve for the other variable.
• Using the equation y = 5 - 2x and substituting x = 16/7:
  • y = 5 - 2(16/7)
  • y = 3/7

Step 5: Verify the Solution

• Substitute both values you found back into both original equations to verify that they are correct.
• In our example, we found x = 16/7 and y = 3/7. Let's check:
  • Equation 1: 2(16/7) + 3/7 = 5 (True)
  • Equation 2: 16/7 - 3(3/7) = 1 (True)

Example

Let's solve the following system of equations using substitution:

• x + 2y = 8
• 3x - y = 1

1. Solve for x in the first equation:
   • x = 8 - 2y
2. Substitute into the second equation:
   • 3(8 - 2y) - y = 1
3. Simplify and solve for y:
   • 24 - 6y - y = 1
   • -7y = -23
   • y = 23/7
4. Substitute y = 23/7 back into the equation x = 8 - 2y:
   • x = 8 - 2(23/7)
   • x = 2/7
5. Verify the solution:
   • Equation 1: 2/7 + 2(23/7) = 8 (True)
   • Equation 2: 3(2/7) - 23/7 = 1 (True)

Therefore, the solution to the system of equations is x = 2/7 and y = 23/7.