Systems of Equations

🔗 Systems of Linear Equations

Part 1 of 7

What Is a System?

A system of equations is a set of two or more equations with the same variables.

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

A solution is an ordered pair $(x, y)$ that satisfies ALL equations simultaneously.

Three Possible Outcomes

Checking a Solution

Is $(3, 1)$ a solution to the system above?

• $2(3)+1 = 7$ ✓
• $3-1 = 2$ ✓

Yes! Both equations are satisfied.

🔄 Substitution Method

Steps:

1. Solve one equation for one variable
2. Substitute into the other equation
3. Solve for the remaining variable
4. Back-substitute to find the other

Example

$\begin{cases} y = 3x - 1 \\ 2x + y = 9 \end{cases}$

Step 1: $y$ is already isolated: $y = 3x - 1$

Step 2: Substitute into equation 2: $2x + (3x-1) = 9$

Step 3: Solve: $5x - 1 = 9 \implies 5x = 10 \implies x = 2$

Step 4: Back-substitute: $y = 3(2)-1 = 5$

Solution: $(2, 5)$

💡 Substitution works best when one variable is already isolated or has coefficient 1.

➕ Elimination Method

Steps:

1. Align equations in standard form ($ax+by=c$)
2. Multiply one or both equations so a variable cancels
3. Add (or subtract) the equations
4. Solve and back-substitute

Example

$\begin{cases} 3x + 2y = 12 \\ 2x - 2y = 8 \end{cases}$

The $y$-terms already cancel when we add:

$5x = 20 \implies x = 4$

Back-substitute: $3(4)+2y = 12 \implies y = 0$.

Solution: $(4, 0)$

When Coefficients Don't Match

$\begin{cases} 3x + 4y = 10 \\ 2x + 3y = 7 \end{cases}$

Multiply eq1 by 3, eq2 by $-4$: $9x+12y = 30$ $-8x-12y = -28$

Add: $x = 2$. Then $y = 1$.

💡 Elimination is ideal when coefficients are already close to matching.

Systems Quiz 🎯

Solve the Systems 🧮

$\begin{cases} 2x+y=10 \\ x-y=2 \end{cases}$

1) $x$ = ?

2) $y$ = ?

3) $\begin{cases} y=4x \\ 2x+y=18 \end{cases}$ → $x$ = ?

Methods & Types 🔽

Exit Quiz ✅

📊 Systems of Three Variables

Part 2 of 7

3×3 Systems

$\begin{cases} x + y + z = 6 \\ 2x - y + z = 3 \\ x + 2y - z = 5 \end{cases}$

A solution is an ordered triple $(x, y, z)$ — a point in 3D space.

Method: Systematic Elimination

1. Choose a variable to eliminate first (pick the easiest)
2. Combine pairs of equations to get TWO equations in TWO variables
3. Solve the 2×2 system
4. Back-substitute to find the third variable

Geometrically

• Each equation represents a plane in 3D
• The solution is where all three planes intersect
• Possibilities: one point, a line, a plane, or no intersection

📝 Worked Example

$\begin{cases} x + y + z = 6 \quad (1) \\ 2x - y + z = 3 \quad (2) \\ x + 2y - z = 5 \quad (3) \end{cases}$

Eliminate $z$: add (1) and (3): $(x+y+z)+(x+2y-z) = 6+5 \implies 2x+3y = 11 \quad (A)$

Add (2) and (3): $(2x-y+z)+(x+2y-z) = 3+5 \implies 3x+y = 8 \quad (B)$

Solve (A) and (B): From (B): $y = 8-3x$. Sub into (A): $2x+3(8-3x)=11 \implies 2x+24-9x=11 \implies -7x=-13 \implies x = \frac{13}{7}$

Hmm, ugly numbers. Let's try a cleaner system for practice.

Cleaner Example

$\begin{cases} x+y+z=6 \\ x-y+z=2 \\ 2x+y-z=1 \end{cases}$

Add eq1+eq2: $2x+2z=8 \implies x+z=4$. Add eq2+eq3: $3x=3 \implies x=1$. Then $z=3$, $y=2$.

Solution: $(1, 2, 3)$ ✓

⚠️ Special Cases in 3D

No Solution (Inconsistent)

$\begin{cases} x+y+z=1 \\ x+y+z=3 \\ 2x+y-z=0 \end{cases}$

Eq1 and eq2 say $x+y+z$ equals both 1 and 3 — contradiction!

Infinitely Many (Dependent)

$\begin{cases} x+y+z=4 \\ 2x+2y+2z=8 \\ x-y+z=0 \end{cases}$

Eq2 = 2×Eq1, so we really have only 2 independent equations in 3 unknowns → infinite solutions (a line).

Application: Curve Fitting

Find the quadratic $y = ax^2+bx+c$ through $(1,6), (2,11), (3,18)$:

$\begin{cases} a+b+c=6 \\ 4a+2b+c=11 \\ 9a+3b+c=18 \end{cases}$

Solving: $a=1, b=2, c=3 \implies y=x^2+2x+3$.

3×3 Systems Quiz 🎯

Solve 🧮

$\begin{cases} x+y+z=10 \\ x-y+z=4 \\ x+y-z=2 \end{cases}$

1) $x$ = ?

2) $y$ = ?

3) $z$ = ?

3D Systems Concepts 🔽

Exit Quiz ✅

📈 Nonlinear Systems

Part 3 of 7

What Are Nonlinear Systems?

At least one equation is not linear (contains $x^2$, $xy$, $\sqrt{x}$, etc.).

$\begin{cases} x^2 + y^2 = 25 \\ x + y = 7 \end{cases}$

Possible Intersections

Main Strategy: Substitution

Nonlinear systems almost always use substitution because elimination may not cancel cleanly.

📝 Line Meets Circle

$\begin{cases} x^2 + y^2 = 25 \\ x + y = 7 \end{cases}$

From eq2: $y = 7-x$. Substitute: $x^2 + (7-x)^2 = 25$ $x^2 + 49 - 14x + x^2 = 25$ $2x^2 - 14x + 24 = 0$ $x^2 - 7x + 12 = 0$ $(x-3)(x-4) = 0$

$x = 3 \implies y = 4$ or $x = 4 \implies y = 3$.

Solutions: $(3, 4)$ and $(4, 3)$.

Line Meets Parabola

$\begin{cases} y = x^2 \\ y = 2x + 3 \end{cases}$

$x^2 = 2x+3 \implies x^2-2x-3=0 \implies (x-3)(x+1)=0$

Solutions: $(3, 9)$ and $(-1, 1)$.

🔬 Two Conics

$\begin{cases} x^2 + y^2 = 10 \\ x^2 - y^2 = 4 \end{cases}$

Add: $2x^2 = 14 \implies x^2 = 7 \implies x = \pm\sqrt{7}$

Subtract: $2y^2 = 6 \implies y^2 = 3 \implies y = \pm\sqrt{3}$

Four solutions: $(\pm\sqrt{7}, \pm\sqrt{3})$ — all four sign combinations!

No-Solution Cases

$\begin{cases} x^2 + y^2 = 1 \\ x^2 + y^2 = 4 \end{cases}$

Concentric circles with different radii — never intersect.

One-Solution (Tangent)

$\begin{cases} y = x^2 \\ y = 2x - 1 \end{cases}$

$x^2 = 2x-1 \implies x^2-2x+1=0 \implies (x-1)^2=0$

Only $x = 1, y = 1$. The line is tangent to the parabola.

Nonlinear Systems Quiz 🎯

Solve 🧮

$\begin{cases} y = x^2 - 1 \\ y = 3x - 1 \end{cases}$

1) Smaller $x$-value: $x$ = ?

2) Larger $x$-value: $x$ = ?

3) Sum of the two $y$-values: $y_1 + y_2$ = ?

Nonlinear Concepts 🔽

Exit Quiz ✅

📊 Systems of Inequalities

Part 4 of 7

Linear Inequalities in Two Variables

$2x + y \leq 6$

The solution is a half-plane — all points on one side of the boundary line.

Graphing Steps

1. Graph the boundary ($=$): solid if $\leq/\geq$, dashed if $</>$
2. Test a point (use $(0,0)$ if not on the line)
3. Shade the side that satisfies the inequality

Example: $2x + y \leq 6$

• Boundary: $y = -2x + 6$ (solid line)
• Test $(0,0)$: $0 + 0 = 0 \leq 6$ ✓ → shade the origin side

Systems of Inequalities

The solution is the intersection of all shaded regions — the overlap.

$\begin{cases} x + y \leq 5 \\ x \geq 0 \\ y \geq 0 \end{cases}$

This gives a triangular region in the first quadrant.

🎯 Feasible Regions & Corner Points

Bounded vs Unbounded

• Bounded: region is enclosed (polygon) — happens when enough constraints
• Unbounded: region extends to infinity

Finding Corner Points

Corner points (vertices) are found by solving pairs of boundary equations simultaneously.

Example: $\begin{cases} x+y \leq 5 \\ 2x+y \leq 8 \\ x \geq 0, y \geq 0 \end{cases}$

Corner points:

• $(0,0)$: intersection of $x=0, y=0$
• $(4,0)$: intersection of $2x+y=8, y=0$
• $(3,2)$: intersection of $x+y=5, 2x+y=8$
• $(0,5)$: intersection of $x+y=5, x=0$

Why Corner Points Matter

Fundamental Theorem of Linear Programming: The max/min of a linear function on a feasible region occurs at a corner point.

💰 Linear Programming

Optimize $P = 3x + 2y$ subject to:

$\begin{cases} x+y \leq 5 \\ 2x+y \leq 8 \\ x \geq 0, y \geq 0 \end{cases}$

Evaluate $P$ at each corner point:

Maximum profit: $P = 13$ at $(3, 2)$.

Real-World Applications

• Manufacturing: maximize profit given resource constraints
• Nutrition: minimize cost while meeting dietary needs
• Scheduling: optimize efficiency under time limits

Inequalities Quiz 🎯

Linear Programming 🧮

Maximize $P = 5x+4y$ with corners $(0,0), (3,0), (2,3), (0,4)$.

1) $P$ at $(2,3)$ = ?

2) $P$ at $(0,4)$ = ?

3) Maximum value of $P$ = ?

Inequality Concepts 🔽

Exit Quiz ✅

🔢 Partial Fractions

Part 5 of 7

Why Partial Fractions?

Decompose complex fractions into simpler ones — essential for integration in calculus!

$\frac{5x+3}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}$

The Process

1. Factor the denominator completely
2. Write one fraction per factor
3. Solve for the unknown constants

Case 1: Distinct Linear Factors

$\frac{5x+3}{(x+1)(x+2)}$

Multiply both sides by $(x+1)(x+2)$: $5x+3 = A(x+2) + B(x+1)$

Method 1 — Strategic substitution:

• Let $x = -1$: $-2 = A(1) \implies A = -2$
• Let $x = -2$: $-7 = B(-1) \implies B = 7$

$\frac{5x+3}{(x+1)(x+2)} = \frac{-2}{x+1} + \frac{7}{x+2}$

📝 Repeated & Quadratic Factors

Case 2: Repeated Linear Factor

$\frac{3x+5}{(x-1)^2} = \frac{A}{x-1} + \frac{B}{(x-1)^2}$

Multiply: $3x+5 = A(x-1)+B$

$x=1$: $8 = B$. Expand: $3x+5 = Ax-A+8 \implies A=3$.

$= \frac{3}{x-1}+\frac{8}{(x-1)^2}$

Case 3: Irreducible Quadratic Factor

$\frac{2x^2+x+3}{(x+1)(x^2+1)} = \frac{A}{x+1}+\frac{Bx+C}{x^2+1}$

Note: quadratic factor gets $Bx+C$ (not just $B$).

$x=-1$: $2-1+3 = A(2) \implies A=2$.

Expand and equate: $B=0, C=1$.

$= \frac{2}{x+1}+\frac{1}{x^2+1}$

🧮 Coefficient Matching Method

When substitution isn't enough, equate coefficients of each power of $x$.

Example

$\frac{x^2+2}{(x-1)(x^2+x+1)} = \frac{A}{x-1}+\frac{Bx+C}{x^2+x+1}$

Multiply: $x^2+2 = A(x^2+x+1)+(Bx+C)(x-1)$

$x=1$: $3 = 3A \implies A = 1$

Expand right side: $x^2+x+1+(Bx^2-Bx+Cx-C)$ $= (1+B)x^2+(1-B+C)x+(1-C)$

Equate coefficients:

• $x^2$: $1 = 1+B \implies B = 0$
• $x^0$: $2 = 1-C \implies C = -1$

$= \frac{1}{x-1}+\frac{-1}{x^2+x+1}$

💡 Always check: is the degree of numerator < degree of denominator? If not, do long division first.

Partial Fractions Quiz 🎯

Find the Constants 🧮

$\frac{7x+1}{(x+1)(x-2)} = \frac{A}{x+1}+\frac{B}{x-2}$

1) $A$ = ? (set $x = -1$)

2) $B$ = ? (set $x = 2$)

3) $\frac{4}{x(x+2)} = \frac{A}{x}+\frac{B}{x+2}$. $A$ = ?

Partial Fractions Concepts 🔽

Exit Quiz ✅

🌍 Applications of Systems

Part 6 of 7

Mixture Problems

Problem: Mix a 30% acid solution with a 70% acid solution to get 100 mL of 40% acid.

Let $x$ = mL of 30%, $y$ = mL of 70%.

$\begin{cases} x + y = 100 \\ 0.30x + 0.70y = 40 \end{cases}$

From eq1: $y = 100-x$. Substitute:

$0.30x + 0.70(100-x) = 40$

$0.30x + 70 - 0.70x = 40$

$-0.40x = -30 \implies x = 75$

Answer: 75 mL of 30% and 25 mL of 70%.

✈️ Distance/Rate/Time Problems

With Wind / Current

Example: Plane flies 600 mi with wind in 2 hrs, return in 3 hrs.

$\begin{cases} (p+w) \cdot 2 = 600 \\ (p-w) \cdot 3 = 600 \end{cases}$

$p+w = 300$ and $p-w = 200$.

Add: $2p = 500 \implies p = 250$ mph. $w = 50$ mph.

Relative Motion

Two trains leave same station in opposite directions at 60 and 80 mph. When are they 350 mi apart?

$60t + 80t = 350 \implies 140t = 350 \implies t = 2.5$ hours.

💰 Investment & Work Problems

Investment

$10,000 split between 5% and 8% annual interest, earning $680 total.

$\begin{cases} x + y = 10000 \\ 0.05x + 0.08y = 680 \end{cases}$

$x = 10000-y$: $0.05(10000-y)+0.08y=680$

$500-0.05y+0.08y=680 \implies 0.03y=180 \implies y=6000$

$4,000 at 5% and $6,000 at 8%.

Work Rate Problems

Worker A: job in 6 hrs. Worker B: job in 4 hrs. Together?

Rates: $\frac{1}{6} + \frac{1}{4} = \frac{2+3}{12} = \frac{5}{12}$

Time together: $\frac{12}{5} = 2.4$ hours.

💡 The key to word problems: define variables clearly and write equations for each constraint.

Applications Quiz 🎯

Word Problems 🧮

1) Sum of two numbers is 20, difference is 6. Larger number = ?

2) 40% + 60% solutions mixed to get 200 mL of 45%. How many mL of 40%?

3) $5,000 at rate $r$ earns $350/yr. Rate (%) = ?

Problem Setup 🔽

Exit Quiz ✅

🏆 Systems of Equations — Full Synthesis

Part 7 of 7

Method Selection Guide

Solution Types Summary

Key Formulas

• Elimination: multiply to match, add/subtract
• Substitution: isolate, plug in, solve
• LP: evaluate objective at corner points
• Partial fractions: factor, decompose, solve for constants

🔄 Mixed Practice Strategies

Quick-Solve Techniques

Symmetric systems: $x+y=S, x-y=D \implies x=\frac{S+D}{2}, y=\frac{S-D}{2}$

Product-Sum: $x+y=s, xy=p$ → solve $t^2-st+p=0$

Three-variable shortcut: Add all equations first to find $x+y+z$.

Example: Mixed Nonlinear

$\begin{cases} x^2+y=10 \\ x+y^2=10 \end{cases}$

By symmetry, try $x=y$: $x^2+x=10 \implies x=\frac{-1+\sqrt{41}}{2} \approx 2.7$

But also check non-symmetric solutions: subtract equations: $x^2-y^2-(x-y)=0 \implies (x-y)(x+y-1)=0$

So either $x=y$ or $x+y=1$ — two families of solutions!

Common Pitfalls

• Forgetting to check solutions in ALL original equations
• Losing solutions when dividing by a variable (might be 0!)
• Not verifying extraneous solutions from squaring

🔗 Calculus Connections

Systems Appear Everywhere

Related Rates (Calculus): Set up systems relating rates of change.

Optimization (Calculus): Lagrange multipliers create systems: $\nabla f = \lambda \nabla g$

Differential Equations: Systems of DEs govern:

• Population dynamics (predator-prey)
• Electrical circuits
• Economic models

Linear Algebra Preview

Systems can be written as matrix equations:

$\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ 2 \end{bmatrix}$

This leads to matrices — our next topic!

Synthesis Quiz 🎯

Final Calculations 🧮

1) $x+y=10, xy=21$. Larger value = ?

2) $\frac{1}{(x-1)(x-2)}$ at $x=3$: value = ?

3) Max of $P=x+2y$ at corners $(0,0),(4,0),(2,3),(0,5)$: max $P$ = ?

Systems Master 🔽

Exit Quiz ✅