System of Linear Equations in Two Variables

Solving Systems of Linear Equations Using Substitution

• Substitution Method Steps:
  • Step 1: Isolate one variable in one equation (preferably the one with the least complex expression).
  • Step 2: Substitute this isolated variable's expression into the other equation.
  • Step 3: Solve the resultant equation for the remaining variable.
  • Step 4: Substitute this found value back to find the value of the other variable using the expression obtained in Step 1.
  • Step 5: Check both equations using the found values.

Example of Substitution

• Given the system:
  • $2x + 3y = 10$
  • $5x - y = -26$

1. Solve for $y$ from the second equation:

   • 5x - y = -26 </li>
   • -y = -5x - 26 </li>
   • y = 5x + 26

2. Substitute $y$ in the first equation:

   • $2x + 3(5x + 26) = 10$
   • $2x + 15x + 78 = 10$
   • $17x + 78 = 10$
   • $17x = 10 - 78$
   • $17x = -68$
   • $x = -4$

3. Substitute $x = -4$ back into the expression for $y$:

   • $y = 5(-4) + 26$
   • $y = -20 + 26$
   • $y = 6$

4. The solution is: (-4, 6).

5. Check:

   • First equation: 2(-4) + 3(6) = 10 </li>
   • -8 + 18 = 10
   • Works!
   • Second equation: $5(-4) - 6 = -26$
   • -20 - 6 = -26
   • Works!

Solving Systems Using Elimination (Addition Method)

• Elimination Method Steps:
  • Step 1: Convert both equations to the form $Ax + By = C$.
  • Step 2: Multiply equations to create coefficient opposites.
  • Step 3: Add the equations to eliminate one variable.
  • Step 4: Solve the new equation for the remaining variable.
  • Step 5: Substitute back into an original equation to find the other variable.
  • Step 6: Check solutions.

Example of Elimination

• Given the system:
  • $2x + 3y = 10$
  • $5x - y = -26$

1. Rewrite as:

   • $2x + 3y = 10$
   • $5x + 1y = -26$

2. Multiply the first equation by 5:

   • $10x + 15y = 50$
   • Multiply the second equation by 3:
   • $15x + 3y = -78$

3. Subtract to eliminate $y$:

   • $10x + 15y - (15x + 3y) = 50 - (-78)$
   • $-5x + 12y = 128$
   • Solve for $x$,
   • $x = -4$.

4. Substitute $x$ into one of the original equations to solve for $y$.

5. Check solutions in both original equations.

Identifying Solutions: Inconsistent or Dependent

• Systems can have:
  • One solution: unique intersection point.
  • No solution: parallel lines - inconsistent.
  • Infinitely many solutions: same line - dependent.

Example Problem

• Given the equations:
  • $10x - 5y = 30$
  • $y = 2x - 6$

1. Substitute $y$:
   • $10x - 5(2x - 6) = 30$
   • Rearrange:
   • $10x - 10x + 30 = 30$
   • True; thus, infinitely many solutions represented as $ext{{(x, y)| y = 2x - 6}}$.

Solving Word Problems Using Systems of Linear Equations

• Example 1: The sum of two numbers is 38. The larger number is 16 more than the smaller.
  • Let larger number = $x$ and smaller = $y$,
  • Equations:
  • $x + y = 38$
  • $x = y + 16$

1. Substitute:
   • $y + 16 + y = 38$
   • Solve: 2y + 16 = 38 </li>
   • 2y = 22 </li>
   • y = 11
   • $x = 27$

Numbers are: 27 and 11.

• Example 2: A fruit company delivers in two sizes.
  • Equations:
  • $2x + 3y = 51$ (weight of boxes)
  • $6x + 5y = 127$

Solution found via substitution or elimination yields:

• Large box = 15.75 kg; Small box = 6.5 kg.

Solving 3x3 Systems of Linear Equations

• General form for three variables $x, y, z$.
• Example equations:
  • $5x - 2y + 2z = -3$
  • $2x + 3y = 51$
  • Additional equation needed for a unique solution.

Determine solutions following similar methods as above:

• Substitution or elimination through [[3 Equations]].
• Check with all original equations.