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Systems of Linear Equations

1️⃣ Understanding Systems of Linear Equations

A system of linear equations is a set of two or more linear equations with the same variables. The goal is to find the values of the variables that satisfy all equations simultaneously.

🔹 General Form (Two Equations, Two Variables):

\[ a_1x + b_1y = c_1 \]

\[ a_2x + b_2y = c_2 \]

  • ✔ \( a_1, b_1, c_1, a_2, b_2, c_2 \) are constants.
  • ✔ \( x \) and \( y \) are the variables we need to solve for.

Example:

\[ 2x + 3y = 8 \quad \text{and} \quad x - y = 2 \]

2️⃣ Graphical Interpretation of a System

Each equation represents a straight line. The solution is the point where the lines intersect.

  1. One Unique Solution: Lines intersect at a single point.
  2. No Solution: Lines are parallel and never intersect.
  3. Infinite Solutions: Lines are identical (overlap completely).

3️⃣ Methods to Solve a System of Equations

Graphing Method

Example: Graph \( y = 2x + 1 \) and \( y = -x + 4 \).
Solution: The lines intersect at \( (1,3) \).

Substitution Method

Example: Solve \( x + 2y = 8 \) and \( x = y + 2 \).
Solution: \( x = 4 \) and \( y = 2 \).

Elimination Method

Example: Solve \( 3x + 2y = 12 \) and \( 2x - 2y = 4 \).
Solution: \( x = \frac{16}{5} \) and \( y = \frac{6}{5} \).

Basic Practice Questions 🎯

  1. Solve using graphing: \( y = 2x + 3 \) and \( y = -x + 5 \).
  2. Solve using substitution: \( x + y = 10 \) and \( x = 3y - 2 \).
  3. Solve using elimination: \( 4x + 3y = 20 \) and \( 2x - 3y = 4 \).
  4. Find the solution for: \( 5x - y = 12 \) and \( x + 2y = 8 \).
  5. Determine if the system \( 3x + 6y = 9 \) and \( x + 2y = 3 \) has one, no, or infinitely many solutions.
  6. A shop sells apples and bananas: 3 apples and 2 bananas cost \$8; 5 apples and 3 bananas cost \$13. Find the cost of one apple and one banana.
  7. A taxi ride: 10 miles cost \$25 and 15 miles cost \$35. Find the fixed charge and per-mile cost.
  8. Solve for \( x \) and \( y \): \( 2x + 5y = 19 \) and \( 3x - y = 4 \).
  9. Find two numbers that sum to 16 and whose difference is 4.
  10. A farm has 50 heads and 140 legs (cows and chickens). How many of each?

Challenging Questions – Test Your Brain! 🤔🔥

  1. A school sells event tickets: student tickets cost \$3, adult tickets cost \$5, and 500 tickets sold total \$2000. Find the number of student and adult tickets.
  2. A chemist mixes a 10% salt solution with a 25% salt solution to make 100 mL of a 20% solution. How much of each is needed?
  3. A theater sells VIP tickets for \$30 and regular tickets for \$15. If 200 tickets sold total \$4500, how many of each were sold?
  4. The sum of two numbers is 50, their difference is 10, and the sum of their squares is 1300. Find the numbers.
  5. The length of a rectangle is three times its width, and the perimeter is 48 cm. Find its dimensions.

5️⃣ Summary

  • ✅ Graphing, Substitution, and Elimination are three methods to solve systems of linear equations.
  • ✅ Systems may have one unique solution, no solution (parallel lines), or infinitely many solutions (identical lines).
  • ✅ Solving systems is useful in many real-life situations like business, finance, and engineering.

6️⃣ Next Topic

Fantastic work! Ready for the next challenge? Let's move on to Word Problems & Real-Life Applications!

Back to Algebra Syllabus