Solving Systems of Equations by Substitution Examples

📚 Quick Study Guide

• 💡 Definition: A system of equations is a set of two or more equations with the same variables. Solving it means finding values for the variables that satisfy all equations simultaneously.
• 🔑 Substitution Method Steps:
  • ➡️ Solve one equation for one variable.
  • 🔄 Substitute that expression into the other equation.
  • 🎯 Solve the resulting equation for the remaining variable.
  • ↩️ Substitute the value back into either original equation to find the value of the other variable.
  • ✅ Check your solution in both original equations.
• ➗ Example: Consider the system: $y = 2x + 1$ $3x + y = 11$ Substitute $2x + 1$ for $y$ in the second equation: $3x + (2x + 1) = 11$ $5x + 1 = 11$ $5x = 10$ $x = 2$ Now, substitute $x = 2$ back into $y = 2x + 1$: $y = 2(2) + 1$ $y = 5$ Solution: $(x, y) = (2, 5)$

Practice Quiz

1. What is the first step in solving a system of equations using the substitution method?
   1. Solve one equation for one variable.
   2. Graph both equations.
   3. Eliminate one variable by adding the equations.
   4. Substitute a random value for one variable.
2. Solve the following system using substitution: $y = x + 1$ $2x + y = 7$
   1. (2, 3)
   2. (1, 2)
   3. (3, 4)
   4. (4, 5)
3. Solve the following system using substitution: $x = 2y$ $x + y = 9$
   1. (6, 3)
   2. (3, 6)
   3. (4, 5)
   4. (9, 0)
4. Which of the following systems is best suited to be solved using the substitution method?
   1. $x + y = 5$ $x - y = 1$
   2. $2x + 3y = 7$ $5x - 3y = 14$
   3. $y = 3x - 2$ $4x + y = 12$
   4. $x + y = 10$ $2x - 2y = 5$
5. Solve the system: $a = 4b - 1$ $2a - 3b = 5$
   1. (7, 2)
   2. (5, 2)
   3. (9, 1)
   4. (1, 0)
6. What does it mean when you substitute and get a false statement, like $0 = 1$?
   1. The system has one solution.
   2. The system has infinitely many solutions.
   3. The system has no solution.
   4. You made a calculation error.
7. Solve the following system using substitution: $p = 5q - 3$ $p = -2q + 11$
   1. (7, 2)
   2. (2, 7)
   3. (3, 12)
   4. (12, 3)