1. What is a System of Equations?

Definition: A system of equations is a set of two or more equations with the same variables. The solution is an ordered pair (x, y) that makes ALL equations true simultaneously.

Example of a System:

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

Key Concepts:

• Solution: An ordered pair (x, y) that satisfies both equations
• Variables: Usually x and y (the unknowns we're solving for)
• System: Must have at least 2 equations

2. Is (x, y) a Solution to the System of Equations?

Steps to Check:

1. Substitute the x-value into BOTH equations
2. Substitute the y-value into BOTH equations
3. Simplify each equation
4. If BOTH equations are true → It IS a solution ✓
5. If either equation is false → It is NOT a solution ✗

Example:

Is (3, 1) a solution to this system?

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

Check Equation 1: \( 2x + y = 7 \)

\( 2(3) + 1 = 7 \) → \( 6 + 1 = 7 \) → \( 7 = 7 \) ✓

Check Equation 2: \( x - y = 2 \)

\( 3 - 1 = 2 \) → \( 2 = 2 \) ✓

Yes, (3, 1) IS a solution because it makes BOTH equations true.

3. Solve a System of Equations by Graphing

Method: Graph both equations on the same coordinate plane. The point where the lines intersect is the solution.

Steps:

1. Write both equations in slope-intercept form: \( y = mx + b \)
2. Graph the first equation (plot y-intercept, use slope)
3. Graph the second equation on the same axes
4. Find the point of intersection
5. Write the solution as an ordered pair (x, y)
6. Check your solution in both original equations

Example:

Solve by graphing:

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

Equation 1: \( y = 2x - 1 \) → slope = 2, y-intercept = -1

Equation 2: \( y = -x + 5 \) → slope = -1, y-intercept = 5

Graph both lines. They intersect at (2, 3)

Solution: (2, 3)

Important Notes:

• The intersection point is the ONLY point that satisfies both equations
• If lines don't intersect (parallel) → No solution
• If lines are the same → Infinitely many solutions

4. Number of Solutions to a System of Equations

Three Possibilities:

How to Determine Without Graphing:

Write both equations in slope-intercept form \( y = mx + b \) and compare:

• Different slopes (m₁ ≠ m₂): One solution
• Same slope, different y-intercepts (m₁ = m₂, b₁ ≠ b₂): No solution
• Same slope, same y-intercept (m₁ = m₂, b₁ = b₂): Infinitely many solutions

Examples:

Example 1: \( y = 3x + 2 \) and \( y = -x + 6 \)

Slopes: 3 and -1 (different) → One solution

Example 2: \( y = 2x + 5 \) and \( y = 2x - 3 \)

Same slope (2), different y-intercepts → No solution (parallel lines)

Example 3: \( y = 4x + 1 \) and \( 2y = 8x + 2 \)

Simplify second: \( y = 4x + 1 \) (same as first) → Infinitely many solutions

5. Classify a System of Equations

Classification Terms:

Consistent: A system that has at least one solution (one solution or infinitely many)

Inconsistent: A system that has no solution (parallel lines)

Independent: A system with exactly one solution (lines intersect at one point)

Dependent: A system with infinitely many solutions (same line)

Complete Classification:

6. Solve a System Using Substitution Method

When to Use: Best when one equation is already solved for a variable, or can be easily solved for a variable.

Steps:

1. Solve one equation for one variable (x or y)
2. Substitute that expression into the other equation
3. Solve for the remaining variable
4. Substitute back to find the other variable
5. Write solution as ordered pair (x, y)
6. Check in both original equations

Example 1:

Solve:

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

Step 1: First equation already solved: \( y = 2x + 1 \)

Step 2: Substitute into second equation:

\( 3x + (2x + 1) = 11 \)

Step 3: Solve for x:

\( 5x + 1 = 11 \) → \( 5x = 10 \) → \( x = 2 \)

Step 4: Substitute x = 2 into \( y = 2x + 1 \):

\( y = 2(2) + 1 = 5 \)

Solution: (2, 5)

Example 2:

Solve:

\( \begin{cases} x + 2y = 10 \\ 3x - y = 1 \end{cases} \)

Step 1: Solve first equation for x: \( x = 10 - 2y \)

Step 2: Substitute into second equation:

\( 3(10 - 2y) - y = 1 \)

Step 3: \( 30 - 6y - y = 1 \) → \( 30 - 7y = 1 \) → \( -7y = -29 \) → \( y = \frac{29}{7} \)

Step 4: \( x = 10 - 2(\frac{29}{7}) = 10 - \frac{58}{7} = \frac{12}{7} \)

Solution: \( (\frac{12}{7}, \frac{29}{7}) \)

7. Solve a System Using Elimination Method

When to Use: Best when variables line up vertically and coefficients can be made opposites easily.

Steps:

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

Example 1: Direct Elimination

Solve:

\( \begin{cases} 2x + y = 8 \\ 3x - y = 7 \end{cases} \)

Step 1: Notice y and -y are already opposites!

Step 2: Add equations:

\( (2x + y) + (3x - y) = 8 + 7 \)

\( 5x = 15 \) → \( x = 3 \)

Step 3: Substitute x = 3 into first equation:

\( 2(3) + y = 8 \) → \( 6 + y = 8 \) → \( y = 2 \)

Solution: (3, 2)

Example 2: Multiply to Create Opposites

Solve:

\( \begin{cases} 3x + 2y = 16 \\ 5x - 2y = 8 \end{cases} \)

Notice: 2y and -2y are already opposites

Add: \( 8x = 24 \) → \( x = 3 \)

Substitute: \( 3(3) + 2y = 16 \) → \( 9 + 2y = 16 \) → \( y = \frac{7}{2} \)

Solution: \( (3, \frac{7}{2}) \)

Example 3: Need to Multiply

Solve:

\( \begin{cases} 2x + 3y = 12 \\ 5x - 4y = 1 \end{cases} \)

Multiply equation 1 by 4: \( 8x + 12y = 48 \)

Multiply equation 2 by 3: \( 15x - 12y = 3 \)

Add: \( 23x = 51 \) → \( x = \frac{51}{23} \)

Then substitute to find y

8. Solve Using Any Method - Which to Choose?

Decision Guide:

Tips for Choosing:

• If \( y = ... \) or \( x = ... \): Use substitution
• If coefficients are already opposites: Use elimination
• If need exact answer with fractions: Avoid graphing
• If unsure: Both algebraic methods work; choose what's easier

9. Systems of Equations: Word Problems

Steps to Solve Word Problems:

1. Read carefully and identify what you're looking for
2. Define variables (let x = ..., let y = ...)
3. Write two equations based on the information given
4. Solve the system using any method
5. Answer the question in a complete sentence with units
6. Check if your answer makes sense

Example 1: Number Problem

Problem: The sum of two numbers is 25. Their difference is 7. Find the numbers.

Variables: Let x = larger number, y = smaller number

Equations:

\( \begin{cases} x + y = 25 \text{ (sum)} \\ x - y = 7 \text{ (difference)} \end{cases} \)

Solution (Elimination): Add equations: \( 2x = 32 \) → \( x = 16 \)

Substitute: \( 16 + y = 25 \) → \( y = 9 \)

Answer: The numbers are 16 and 9.

Example 2: Money Problem

Problem: Adult tickets cost $8 and child tickets cost $5. A total of 100 tickets were sold for $650. How many of each type were sold?

Variables: Let a = adult tickets, c = child tickets

Equations:

\( \begin{cases} a + c = 100 \text{ (total tickets)} \\ 8a + 5c = 650 \text{ (total money)} \end{cases} \)

Solution (Substitution): From equation 1: \( c = 100 - a \)

Substitute into equation 2: \( 8a + 5(100 - a) = 650 \)

\( 8a + 500 - 5a = 650 \) → \( 3a = 150 \) → \( a = 50 \)

\( c = 100 - 50 = 50 \)

Answer: 50 adult tickets and 50 child tickets were sold.

Example 3: Mixture Problem

Problem: A solution is made by mixing 20% acid solution with 50% acid solution. How many liters of each are needed to make 30 liters of 35% acid solution?

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

Equations:

\( \begin{cases} x + y = 30 \text{ (total volume)} \\ 0.20x + 0.50y = 0.35(30) \text{ (acid amount)} \end{cases} \)

Simplify equation 2: \( 0.20x + 0.50y = 10.5 \)

Solve to find x = 15 liters and y = 15 liters

Quick Reference: Systems of Equations

Three Methods:

1. Graphing: Graph both lines, find intersection point

2. Substitution: Solve one equation for a variable, substitute into other

3. Elimination: Add/subtract equations to eliminate a variable

Number of Solutions:

• One solution: Lines intersect (different slopes)
• No solution: Parallel lines (same slope, different y-intercepts)
• Infinitely many: Same line (same slope and y-intercept)

Classification:

• Consistent: Has at least one solution
• Inconsistent: Has no solution
• Independent: Exactly one solution
• Dependent: Infinitely many solutions

💡 Key Tips for Systems of Equations

• ✓ Solution must satisfy BOTH equations
• ✓ Always check your answer in both original equations
• ✓ Graphing: intersection point is the solution
• ✓ Substitution: best when one variable is isolated
• ✓ Elimination: make coefficients opposites, then add
• ✓ Parallel lines = no solution = inconsistent
• ✓ Same line = infinite solutions = dependent
• ✓ Different slopes = one solution = independent
• ✓ Word problems: define variables clearly first
• ✓ Look for key words: sum, difference, total, each
• ✓ Write two separate equations from given info
• ✓ Answer word problems with complete sentences + units