Steps to Check if a Point is a Solution:

1. Step 1: Substitute the x and y values into the first equation
2. Step 2: Check if the equation is true (LHS = RHS)
3. Step 3: Substitute the same values into the second equation
4. Step 4: Check if this equation is also true
5. Result: The point is a solution ONLY if it satisfies ALL equations

Example: Is (3, 1) a solution?

System: \( \begin{cases} 2x + y = 7 \\ x - y = 2 \end{cases} \)

Check Equation 1: \( 2(3) + 1 = 6 + 1 = 7 \) ✓ True

Check Equation 2: \( 3 - 1 = 2 \) ✓ True

Answer: Yes, (3, 1) is a solution because it satisfies both equations

Steps to Solve by Graphing:

1. Step 1: Write both equations in slope-intercept form \( y = mx + b \)
2. Step 2: Graph both lines on the same coordinate plane
3. Step 3: Find the point of intersection
4. Step 4: The coordinates of the intersection point are the solution

Three Possible Cases:

• One Solution: Lines intersect at one point (different slopes)
• No Solution: Lines are parallel (same slope, different y-intercepts)
• Infinite Solutions: Lines coincide (same slope, same y-intercept)

Example: Solve by Graphing

System: \( \begin{cases} y = 2x - 1 \\ y = -x + 5 \end{cases} \)

Step 1: Both already in slope-intercept form

Step 2: Graph both lines (Line 1: slope=2, y-int=-1; Line 2: slope=-1, y-int=5)

Step 3: Lines intersect at (2, 3)

Solution: \( x = 2, y = 3 \) or (2, 3)

Steps for Word Problems:

1. Read and understand the problem
2. Define variables for unknowns
3. Write two equations based on the given information
4. Graph both equations
5. Find the intersection point
6. Interpret the solution in the context of the problem

Example: Cost Problem

"Company A charges $50 plus $2 per mile. Company B charges $30 plus $3 per mile. For what mileage do they cost the same?"

Let: x = miles, y = total cost

Company A: \( y = 2x + 50 \)

Company B: \( y = 3x + 30 \)

Graph and find intersection: (20, 90)

Answer: At 20 miles, both cost $90

Determining Number of Solutions:

Examples:

1. \( y = 2x + 3 \) and \( y = -x + 1 \) → Different slopes → One solution

2. \( y = 3x + 2 \) and \( y = 3x - 5 \) → Same slope, different y-int → No solution

3. \( 2x + y = 4 \) and \( 4x + 2y = 8 \) → Same line → Infinite solutions

Classification Terms:

• Consistent: Has at least one solution (one or infinite)
  • Independent: Exactly one solution
  • Dependent: Infinitely many solutions
• Inconsistent: Has no solution (parallel lines)

Quick Classification Guide:

Consistent & Independent: Different slopes → Lines intersect once

Consistent & Dependent: Identical equations → Same line

Inconsistent: Same slope, different intercepts → Parallel lines

Steps for Substitution Method:

1. Step 1: Solve one equation for one variable (choose the easiest)
2. Step 2: Substitute this expression into the other equation
3. Step 3: Solve the resulting equation for the remaining variable
4. Step 4: Substitute back to find the other variable
5. Step 5: Check your solution in both original equations

Example: Solve by Substitution

System: \( \begin{cases} y = 2x + 1 \\ 3x + y = 11 \end{cases} \)

Step 1: First equation already solved for y

Step 2: Substitute into second: \( 3x + (2x + 1) = 11 \)

Step 3: Solve: \( 5x + 1 = 11 \) → \( 5x = 10 \) → \( x = 2 \)

Step 4: Find y: \( y = 2(2) + 1 = 5 \)

Solution: (2, 5)

Example: Age Problem

"Sarah is 3 years older than twice Tom's age. The sum of their ages is 39. How old is each person?"

Let: s = Sarah's age, t = Tom's age

Equation 1: \( s = 2t + 3 \)

Equation 2: \( s + t = 39 \)

Substitute: \( (2t + 3) + t = 39 \)

Solve: \( 3t + 3 = 39 \) → \( 3t = 36 \) → \( t = 12 \)

Find s: \( s = 2(12) + 3 = 27 \)

Answer: Tom is 12 years old, Sarah is 27 years old

Steps for Elimination Method:

1. Step 1: Write both equations in standard form \( Ax + By = C \)
2. Step 2: Multiply one or both equations to make coefficients of one variable opposites
3. Step 3: Add the equations to eliminate one variable
4. Step 4: Solve for the remaining variable
5. Step 5: Substitute back to find the other variable
6. Step 6: Check your solution

Example 1: Simple Elimination

System: \( \begin{cases} 2x + y = 10 \\ 3x - y = 5 \end{cases} \)

Step 1: Equations already in standard form

Step 2: Coefficients of y are already opposites (+1 and -1)

Step 3: Add equations: \( 5x = 15 \)

Step 4: \( x = 3 \)

Step 5: Substitute: \( 2(3) + y = 10 \) → \( y = 4 \)

Solution: (3, 4)

Example 2: Multiplication Required

System: \( \begin{cases} 2x + 3y = 12 \\ 5x - 2y = 11 \end{cases} \)

Step 1: Multiply first by 2: \( 4x + 6y = 24 \)

Step 2: Multiply second by 3: \( 15x - 6y = 33 \)

Step 3: Add equations: \( 19x = 57 \) → \( x = 3 \)

Step 4: Substitute: \( 2(3) + 3y = 12 \) → \( 3y = 6 \) → \( y = 2 \)

Solution: (3, 2)

Example: Ticket Sales

"Adult tickets cost $8 and child tickets cost $5. If 150 tickets were sold for $1020, how many of each type were sold?"

Let: a = adult tickets, c = child tickets

Equation 1: \( a + c = 150 \) (total tickets)

Equation 2: \( 8a + 5c = 1020 \) (total money)

Multiply Eq 1 by -5: \( -5a - 5c = -750 \)

Add to Eq 2: \( 3a = 270 \) → \( a = 90 \)

Find c: \( 90 + c = 150 \) → \( c = 60 \)

Answer: 90 adult tickets, 60 child tickets

Choosing the Best Method:

• Use Graphing when: You need a visual representation or approximate solution
• Use Substitution when: One variable is already isolated or easy to isolate
• Use Elimination when: Coefficients are convenient or both equations are in standard form

Decision Guide:

Best for Substitution: \( y = 3x - 5 \) and \( 2x + y = 10 \) (y already isolated)

Best for Elimination: \( 3x + 2y = 12 \) and \( 3x - 5y = 6 \) (same x coefficient)

Either works: Choose the method you're most comfortable with!

Example: Mixture Problem

"A solution is 20% acid and another is 50% acid. How many liters of each are needed to make 30 liters of 35% acid solution?"

Let: x = liters of 20% solution, y = liters of 50% solution

Equation 1: \( x + y = 30 \) (total volume)

Equation 2: \( 0.20x + 0.50y = 0.35(30) \) → \( 0.20x + 0.50y = 10.5 \)

Solve: Using substitution or elimination → \( x = 15, y = 15 \)

Answer: 15 liters of each solution

Standard Form for 3 Variables:

\( \begin{cases} ax + by + cz = d_1 \\ ex + fy + gz = d_2 \\ hx + iy + jz = d_3 \end{cases} \)

Steps for 3-Variable Substitution:

1. Solve one equation for one variable
2. Substitute into the other two equations
3. Solve the resulting 2-variable system
4. Back-substitute to find all three variables
5. Write as ordered triple (x, y, z)

Example:

System: \( \begin{cases} x + y + z = 6 \\ 2x - y + z = 3 \\ x + 2y - z = 4 \end{cases} \)

From Eq 1: \( z = 6 - x - y \)

Substitute into Eq 2 & 3, solve system

Solution: (1, 2, 3)

Steps for 3-Variable Elimination:

1. Step 1: Choose a variable to eliminate
2. Step 2: Use two pairs of equations to eliminate the chosen variable
   • Eliminate from Equations 1 & 2
   • Eliminate from Equations 1 & 3 (or 2 & 3)
3. Step 3: You now have 2 equations with 2 variables
4. Step 4: Solve this 2-variable system
5. Step 5: Substitute back to find the third variable
6. Step 6: Check solution in all three original equations

Example:

System: \( \begin{cases} x + y + z = 9 \\ 2x - y + 3z = 14 \\ x + 2y - z = 5 \end{cases} \)

Eliminate y: Add Eq 1 & 2 → \( 3x + 4z = 23 \)

Eliminate y: Multiply Eq 1 by 2, subtract Eq 3 → \( x + 3z = 13 \)

Solve 2-variable system: \( x = 1, z = 4 \)

Find y: \( 1 + y + 4 = 9 \) → \( y = 4 \)

Solution: (1, 4, 4)

Three Possible Cases:

How to Determine:

• One Solution: Elimination/substitution yields unique values for x, y, and z
• No Solution: Process leads to contradiction (e.g., 0 = 3)
• Infinite Solutions: Process leads to identity (e.g., 0 = 0), equations are dependent

Solution Methods:

• Graphing: Plot both lines, solution is intersection point
• Substitution: Solve for one variable, substitute into other equation
• Elimination: Add/subtract equations to eliminate one variable
• Three Variables: Eliminate to get 2-variable system, then solve