Linear Equations: Substitution and Elimination Methods

Linear Equations

Substitution Method

• A method used to solve systems of linear equations by substituting one equation into another
• Steps:
  1. Choose one of the equations and solve for one variable (e.g. x or y)
  2. Substitute the expression from step 1 into the other equation
  3. Simplify and solve for the other variable
  4. Substitute the value of the variable back into one of the original equations to find the value of the other variable

Elimination Method

• A method used to solve systems of linear equations by eliminating one variable
• Steps:
  1. Multiply both equations by necessary multiples such that the coefficients of one variable (e.g. x or y) are the same
  2. Add or subtract the equations to eliminate the variable
  3. Simplify and solve for the remaining variable
  4. Substitute the value of the variable back into one of the original equations to find the value of the other variable

Key Points

• Both methods can be used to solve systems of linear equations
• The substitution method is often used when one equation is easily solvable for one variable
• The elimination method is often used when the coefficients of one variable are easily made equal
• Graphing can also be used to solve systems of linear equations, but is not always the most efficient method

Linear Equations Solutions

Substitution Method

• Solves systems of linear equations by substituting one equation into another
• Steps involve choosing an equation, solving for one variable, substituting into the other equation, simplifying, and solving for the other variable
• Involves back-substitution to find the value of the other variable

Elimination Method

• Solves systems of linear equations by eliminating one variable
• Steps involve multiplying equations to make coefficients equal, adding or subtracting to eliminate a variable, simplifying, and solving for the remaining variable
• Involves back-substitution to find the value of the other variable

Key Points

• Both substitution and elimination methods can be used to solve systems of linear equations
• Substitution method is suitable when one equation is easily solvable for one variable
• Elimination method is suitable when coefficients of one variable can be easily made equal
• Graphing can also be used to solve systems of linear equations, but may not always be the most efficient method

Nutrition

• Autotrophic nutrition: organisms produce their own food, e.g., plants
• Heterotrophic nutrition: organisms cannot produce their own food, e.g., animals

Modes of Nutrition

• Holozoic nutrition: ingestion of solid food particles, e.g., animals
• Saprotrophic nutrition: absorption of nutrients from dead organic matter, e.g., fungi
• Parasitic nutrition: obtaining nutrients from another living organism, e.g., parasites

Photosynthesis

• Definition: process by which plants, algae, and some bacteria convert light energy into chemical energy
• Overall equation: 6 CO2 + 6 H2O + light energy → C6H12O6 (glucose) + 6 O2
• Light-dependent reactions: occur in thylakoid membranes, light energy excites electrons to generate ATP and NADPH
• Light-independent reactions (Calvin Cycle): occur in stroma, ATP and NADPH are used to convert CO2 into glucose

Respiration

• Definition: process by which cells generate energy from glucose
• Types of respiration:
  • Aerobic respiration: occurs in presence of oxygen, produces ATP and water
  • Anaerobic respiration: occurs in absence of oxygen, produces ATP and lactic acid or ethanol

Stages of Aerobic Respiration

• Glycolysis: breakdown of glucose into pyruvate
• Citric Acid Cycle (Krebs Cycle): breakdown of pyruvate into ATP, NADH, and FADH2
• Electron Transport Chain: generation of ATP from NADH and FADH2
• Oxidative Phosphorylation: production of ATP from energy released during electron transport