Understanding Simultaneous Equations

Introduction to Simultaneous Equations

Definition and Importance

• Simultaneous Equations: Groups of equations involving multiple variables, which are solved together to determine a common solution.
• Analogy: Similar to assembling a jigsaw puzzle, each equation integrates to unveil the entire solution.
• Importance:
  • Essential for comprehending interconnections between variables.
  • Extensively applied in physics, engineering, and computer science for real-world modelling.

Types of Simultaneous Equations

• Linear Equations:

  • Form: $ax + by = c$.
  • Example: Solve $2x + 3y = 6$ and $x - y = 1$.

• Quadratic Equations:

  • Form: $ax^2 + bx + c = 0$.
  • Method: Factoring or using the Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

• Cubic Equations:

  • Includes terms such as $ax^3$. Mentioned for comprehensiveness in understanding complexity.

Key Terminology Introduction

• Solution Set:

  • All feasible solutions that satisfy the equations.
  • Example: For $x^2 - 1 = 0$, solution set: $\{1, -1\}$.

• Consistency:

  • Consistent System: Contains at least one solution, e.g., $x + y = 2$.
  • Inconsistent System: Lacks a solution, e.g., $x + y = 2$ and $x + y = 3$.

• Intersection:

  • Graphical location where equations meet signifies solutions.

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Solving Methods

Substitution Method

Step-by-Step Guide

• Step 1: Isolate one variable in an equation.
• Step 2: Substitute this expression in the other equation.
• Step 3: Solve for the second variable.
• Step 4: Verify the solution by re-substituting it into the original equation.
• Example: Solve $y = 2x + 3$ and $3x + y = 11$.

Solution: From the first equation, $y = 2x + 3$. Substituting into the second equation: $3x + (2x + 3) = 11$ $3x + 2x + 3 = 11$ $5x + 3 = 11$ $5x = 8$ $x = \frac{8}{5}$

Now find $y$ by substituting $x = \frac{8}{5}$ into $y = 2x + 3$: $y = 2(\frac{8}{5}) + 3$ $y = \frac{16}{5} + 3$ $y = \frac{16}{5} + \frac{15}{5} = \frac{31}{5}$

Therefore, the solution is $x = \frac{8}{5}$ and $y = \frac{31}{5}$.

Elimination Method

Step-by-Step Guide

• Step 1: Adjust equations to equalise coefficients.
• Step 2: Add or subtract equations to eliminate a variable.
• Step 3: Solve the resultant equation for one variable.
• Step 4: Utilise this solution to resolve the second variable.
• Example: Solve $2x + 3y = 9$ and $4x - 3y = 15$.

Solution: Add the two equations to eliminate $y$: $(2x + 3y) + (4x - 3y) = 9 + 15$ $6x = 24$ $x = 4$

Substitute $x = 4$ into the first equation: $2(4) + 3y = 9$ $8 + 3y = 9$ $3y = 1$ $y = \frac{1}{3}$

Therefore, the solution is $x = 4$ and $y = \frac{1}{3}$.

Matrix Method

Overview and Operations

• Matrix: An organised array of numbers in rows and columns.

• Steps:
  • Convert equations into matrix format.
  • Implement Gaussian elimination or matrix inversion.
  • Confirm results using computational tools.

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Graphical Representation of Solutions

Graphical Method

• Graphical Solution: Equations are graphed to locate intersections as solutions.
• Steps to Successful Graphing:
  1. Graph each line based on equations.
  2. Identify where the lines intersect.
  3. Intersection points denote solutions.

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Interpretation of Solutions

Solution Analysis

• Unique Solution: Depicted as lines intersecting at one point. Key for accurate scheduling.
• No Solution: Indicated by parallel lines.
• Infinite Solutions: Represented by overlapping lines.

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Real-World Applications Overview

Simultaneous equations are essential in fields like business, economics, and the sciences. They convert complex mathematical concepts into practical solutions.

Examples

Business Application: Break-Even Analysis

• Example:
  • Cost Equation: $C(x) = 50 + 5x$
  • Revenue Equation: $R(x) = 10x$
  • Break-Even Point: Solve $50 + 5x = 10x$ resulting in $x = 10$ units.

Economic Application: Market Equilibrium

• Numerical Example:
  • Supply Equation: $S(p) = 2p + 10$
  • Demand Equation: $D(p) = 50 - 3p$
  • Equilibrium Price: Solve $2p + 10 = 50 - 3p$, resulting in $p = 8$.

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Exam Tips

• Verification: Consistently re-evaluate and confirm solutions.
• Graphical Understanding: Visual tools aid understanding.
• Technique Choice: Match the method to the complexity of the problem.

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Practice Problems with Solutions

1. Solve $x + 2y = 8$ and $3x - y = 7$ using substitution.

Solution: From the first equation, isolate $x$: $x = 8 - 2y$

Substitute into the second equation: $3(8 - 2y) - y = 7$ $24 - 6y - y = 7$ $24 - 7y = 7$ $-7y = -17$ $y = \frac{17}{7}$

Now find $x$ by substituting $y = \frac{17}{7}$ into $x = 8 - 2y$: $x = 8 - 2(\frac{17}{7}) = 8 - \frac{34}{7} = \frac{56 - 34}{7} = \frac{22}{7}$

Therefore, the solution is $x = \frac{22}{7}$ and $y = \frac{17}{7}$.

2. Solve using elimination: $3x + 2y = 16$, $5x + 2y = 24$.

Solution: Subtract the first equation from the second: $(5x + 2y) - (3x + 2y) = 24 - 16$ $2x = 8$ $x = 4$

Substitute $x = 4$ into the first equation: $3(4) + 2y = 16$ $12 + 2y = 16$ $2y = 4$ $y = 2$

Therefore, the solution is $x = 4$ and $y = 2$.