🔟Elementary Algebra Unit 5 – Systems of Linear Equations

Key Concepts

• Linear equations involve variables raised to the first power and can be written in the form $ax + by = c$
• The solution to a linear equation is an ordered pair $(x, y)$ that satisfies the equation
• Slope measures the steepness of a line and can be calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$
  • Slope is represented by the letter $m$ in the slope-intercept form of a linear equation $y = mx + b$
• The $y$-intercept is the point where a line crosses the $y$-axis and can be found by setting $x = 0$ in the equation
• Parallel lines have the same slope but different $y$-intercepts
• Perpendicular lines have slopes that are negative reciprocals of each other

Types of Linear Equations

• Slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept
• Point-slope form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope
• Standard form: $ax + by = c$, where $a$, $b$, and $c$ are constants and $a$ and $b$ are not both zero
  • To convert from standard form to slope-intercept form, solve for $y$
• Vertical lines have equations in the form $x = k$, where $k$ is a constant
• Horizontal lines have equations in the form $y = k$, where $k$ is a constant

Solving Linear Equations

• To solve a linear equation, isolate the variable on one side of the equation using inverse operations
  • Add or subtract the same value from both sides of the equation to eliminate constants
  • Multiply or divide both sides of the equation by the same non-zero value to eliminate coefficients
• Check your solution by substituting the value back into the original equation
• Equations with no solution (inconsistent) result in a false statement like $0 = 1$ when solved
• Equations with infinitely many solutions (dependent) result in a true statement like $0 = 0$ when solved

Graphing Linear Equations

• To graph a linear equation, find at least two points that satisfy the equation and plot them on the coordinate plane
  • Connect the points with a straight line using a ruler
• The $x$-intercept is the point where a line crosses the $x$-axis and can be found by setting $y = 0$ in the equation
• Intercept form of a linear equation: $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ and $b$ are the $x$ and $y$ intercepts, respectively
• Horizontal lines are always parallel to the $x$-axis and have a slope of zero
• Vertical lines are always perpendicular to the $x$-axis and have an undefined slope

Systems of Linear Equations

• A system of linear equations consists of two or more linear equations with the same variables
• The solution to a system of linear equations is an ordered pair $(x, y)$ that satisfies all equations in the system simultaneously
• Systems can have one unique solution, no solution, or infinitely many solutions
  • One solution: The lines intersect at a single point (consistent and independent)
  • No solution: The lines are parallel and do not intersect (inconsistent)
  • Infinitely many solutions: The lines coincide and have the same equation (consistent and dependent)

Solution Methods

• Graphing: Plot both equations on the same coordinate plane and find the point of intersection, if any
  • This method is useful for visualizing the solution but may not always provide an exact answer
• Substitution: Solve one equation for a variable and substitute the resulting expression into the other equation
  • Solve the resulting equation for the remaining variable and substitute back to find the other value
• Elimination: Multiply one or both equations by constants to make the coefficients of one variable equal in magnitude but opposite in sign
  • Add the equations to eliminate one variable, then solve for the remaining variable and substitute back

Applications and Word Problems

• Identify the unknown quantities and assign variables to represent them
• Write a linear equation (or system of equations) based on the given information and relationships between the variables
  • Example: If the sum of two numbers is 50 and their difference is 14, let $x$ be the smaller number and $y$ be the larger number
    • Equation 1: $x + y = 50$
    • Equation 2: $y - x = 14$
• Solve the equation(s) using an appropriate method and interpret the solution in the context of the problem
• Check if the solution makes sense and satisfies any given constraints

Common Mistakes and Tips

• Make sure to distribute negative signs correctly when expanding or simplifying equations
• Be careful when multiplying or dividing both sides of an equation by a variable term, as this may introduce extraneous solutions
  • Always check your solutions in the original equation to verify their validity
• When solving systems by elimination, make sure to multiply equations by constants that will result in coefficients with equal magnitude but opposite signs
• Pay attention to the context and units of the variables in word problems to ensure your solution is meaningful
• Double-check your algebra and calculations to avoid simple arithmetic errors
• Practice solving a variety of problems to develop proficiency and understanding of the concepts