5.2 Solving Systems of Equations by Substitution

Solving systems of equations by substitution is a key technique in algebra. It involves isolating a variable in one equation and plugging it into another, allowing you to solve for both variables step-by-step.

This method is particularly useful when one equation is already solved for a variable. It's often more efficient than other methods, especially with complex fractions or when one variable has a coefficient of 1 or -1.

Solving Systems of Equations by Substitution

Substitution method for linear systems

• Involves solving one equation for one variable and substituting the resulting expression into the other equation
  • Isolate one variable in one of the equations (choose the equation and variable resulting in the simplest algebraic expression)
  • Substitute this expression for the variable into the other equation
  • Solve the resulting equation for the remaining variable
  • Substitute the value of the solved variable back into the expression from the first step to find the value of the other variable
  • Check the solution by substituting the values of both variables into the original equations ensuring they satisfy both equations ($x = 2$, $y = 3$)

Real-world applications of substitution

• Identify variables in the problem and assign them to appropriate quantities (let $x$ be the number of apples, let $y$ be the number of oranges)
• Create a system of equations based on given information and relationships between variables
  • Ensure equations are linear with two variables ($2x + 3y = 10$, $x - y = 1$)
• Use substitution method to solve the system of equations
  • Follow steps outlined in the substitution method for linear systems
• Interpret the solution in the context of the original problem
  • Ensure the solution makes sense and answers the question posed (the store sold 3 apples and 2 oranges)

Efficiency of substitution method

• Often most efficient when one equation has a variable with a coefficient of 1 or -1
  • Allows for easy isolation of the variable, making substitution straightforward ($y = 2x + 1$)
• Preferred when one equation is already solved for one of the variables
  • Eliminates need for the first step of the substitution process ($y = 3x - 2$)
• When coefficients of variables in both equations are large or complex fractions, substitution may be easier than other methods like elimination
  • Avoids need to multiply equations by constants to eliminate a variable, which can lead to more complex calculations ($\frac{2}{3}x + \frac{1}{2}y = 5$, $\frac{1}{4}x - \frac{3}{5}y = 2$)

Understanding Systems of Equations

• An equation is a mathematical statement that two expressions are equal
• A system of equations, also known as simultaneous equations, consists of two or more equations that are considered together
• The solution set of a system is the set of all ordered pairs that satisfy all equations in the system