Systems of Equations
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Solve a System of Equations by Graphing
Method:
The solution to a system is the point where the graphs intersect
Steps:
1. Write each equation in slope-intercept form \( y = mx + b \)
2. Graph both lines on the same coordinate plane
3. Identify the point of intersection (x, y)
4. Verify the solution in both original equations
Example:
Solve by graphing:
\[ \begin{cases} y = 2x - 1 \\ y = -x + 5 \end{cases} \]
Both equations already in slope-intercept form
Graph both lines and find intersection point
Solution: (2, 3)
2. Classify a System of Equations
Types of Systems:
Consistent and Independent:
• Lines intersect at exactly ONE point
• Has one unique solution
• Different slopes: \( m_1 \neq m_2 \)
Consistent and Dependent:
• Same line (equations are equivalent)
• Infinitely many solutions
• Same slope and y-intercept: \( m_1 = m_2 \) and \( b_1 = b_2 \)
Inconsistent:
• Parallel lines (never intersect)
• No solution
• Same slope, different y-intercepts: \( m_1 = m_2 \) but \( b_1 \neq b_2 \)
Classification Table:
| Type | Graphical View | Number of Solutions |
|---|---|---|
| Consistent Independent | Lines intersect at one point | Exactly one |
| Consistent Dependent | Same line (overlap) | Infinitely many |
| Inconsistent | Parallel lines | None |
3. Solve Using Substitution
Steps:
1. Solve one equation for one variable (choose easiest)
2. Substitute that expression into the other equation
3. Solve for the remaining variable
4. Substitute back to find the other variable
Example:
Solve using substitution:
\[ \begin{cases} x + 2y = 7 \\ 3x - y = 5 \end{cases} \]
Solve first equation for x: \( x = 7 - 2y \)
Substitute into second: \( 3(7 - 2y) - y = 5 \)
Expand: \( 21 - 6y - y = 5 \)
Simplify: \( 21 - 7y = 5 \) → \( -7y = -16 \) → \( y = \frac{16}{7} \)
Substitute back: \( x = 7 - 2(\frac{16}{7}) = \frac{17}{7} \)
Solution: \( \left(\frac{17}{7}, \frac{16}{7}\right) \)
4. Solve Using Elimination
Steps:
1. Arrange equations in standard form \( ax + by = c \)
2. Multiply one or both equations to make coefficients opposite
3. Add or subtract equations to eliminate one variable
4. Solve for the remaining variable
5. Substitute back to find the other variable
Example:
Solve using elimination:
\[ \begin{cases} 2x + 3y = 8 \\ 3x - 3y = 12 \end{cases} \]
Notice: y-coefficients are already opposites (3 and -3)
Add equations: \( (2x + 3y) + (3x - 3y) = 8 + 12 \)
Simplify: \( 5x = 20 \) → \( x = 4 \)
Substitute x = 4 into first equation: \( 2(4) + 3y = 8 \)
Solve: \( 8 + 3y = 8 \) → \( y = 0 \)
Solution: (4, 0)
5. Solve Using Augmented Matrices
Augmented Matrix Form:
For system \( \begin{cases} ax + by = e \\ cx + dy = f \end{cases} \), write as:
\[ \left[\begin{array}{cc|c} a & b & e \\ c & d & f \end{array}\right] \]
Row Operations:
1. Swap rows: \( R_i \leftrightarrow R_j \)
2. Multiply row by constant: \( kR_i \)
3. Add multiple of one row to another: \( R_i + kR_j \)
Goal: Row-Echelon Form:
\[ \left[\begin{array}{cc|c} 1 & a & b \\ 0 & 1 & c \end{array}\right] \]
Ones on diagonal, zeros below
Example:
Solve: \( \begin{cases} 2x + y = 5 \\ x - y = 1 \end{cases} \)
Write augmented matrix: \( \left[\begin{array}{cc|c} 2 & 1 & 5 \\ 1 & -1 & 1 \end{array}\right] \)
\( R_1 \leftrightarrow R_2 \): \( \left[\begin{array}{cc|c} 1 & -1 & 1 \\ 2 & 1 & 5 \end{array}\right] \)
\( R_2 - 2R_1 \): \( \left[\begin{array}{cc|c} 1 & -1 & 1 \\ 0 & 3 & 3 \end{array}\right] \)
\( \frac{1}{3}R_2 \): \( \left[\begin{array}{cc|c} 1 & -1 & 1 \\ 0 & 1 & 1 \end{array}\right] \)
Read solution: \( y = 1 \), \( x - 1 = 1 \) → \( x = 2 \)
Solution: (2, 1)
6. Three Variables Using Substitution
General Form:
\[ \begin{cases} ax + by + cz = d \\ ex + fy + gz = h \\ ix + jy + kz = m \end{cases} \]
Steps:
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
7. Three Variables Using Elimination
Strategy:
1. Choose a variable to eliminate
2. Use pairs of equations to eliminate that variable twice
3. Solve the resulting 2-variable system
4. Back-substitute to find all three variables
Example:
Solve:
\[ \begin{cases} x + y + z = 6 \\ 2x - y + z = 3 \\ x + 2y - z = 0 \end{cases} \]
Add equations 1 and 3 to eliminate z: \( 2x + 3y = 6 \)
Add equations 2 and 3 to eliminate z: \( 3x + y = 3 \)
Now solve 2-variable system for x and y
Then substitute back to find z
Solution: (1, \frac{4}{3}, \frac{11}{3})
8. Number of Solutions (Three Variables)
Three Possibilities:
One Unique Solution:
• Three planes intersect at exactly one point
• System is consistent and independent
• Ordered triple (x, y, z)
Infinitely Many Solutions:
• Three planes coincide (same plane)
• Or three planes intersect along a line
• System is consistent and dependent
• Solution expressed with parameters
No Solution:
• Planes don't all intersect at common point
• At least one pair is parallel or contradictory
• System is inconsistent
How to Determine:
• If you get a specific value for each variable → One solution
• If you get an identity (0 = 0) → Infinitely many solutions
• If you get a contradiction (0 = 5) → No solution
9. Word Problem Strategies
Steps for Word Problems:
1. Define variables: What are you solving for?
2. Write equations: Translate words into math
3. Solve the system: Use graphing, substitution, elimination, or matrices
4. Check and interpret: Does the answer make sense?
Common Types:
• Mixture problems (concentrations, solutions)
• Distance/rate/time problems
• Investment/interest problems
• Number problems
• Cost/revenue problems
10. Quick Reference Summary
Method Comparison:
| Method | Best When... |
|---|---|
| Graphing | Visual understanding needed, approximate solution OK |
| Substitution | One variable already isolated or easy to isolate |
| Elimination | Coefficients line up well for adding/subtracting |
| Matrices | Large systems, calculator available, organized approach needed |
📚 Study Tips
✓ Always check your solution in ALL original equations
✓ Substitution is best when one variable is already isolated
✓ Elimination is best when coefficients match or are easy to match
✓ Parallel lines mean no solution; same line means infinite solutions
✓ For 3 variables, eliminate same variable from two pairs of equations