5.1 Solve Systems of Equations by Graphing

Solving systems of equations by graphing is a powerful visual method for finding solutions. It involves plotting equations on a coordinate plane and identifying where the lines intersect. This approach helps students understand the relationship between equations and their graphical representations.

Graphing systems of equations allows us to see how many solutions exist and where they occur. By plotting lines and finding intersection points, we can determine if a system has one solution, no solution, or infinitely many solutions. This visual method connects algebra to geometry in a practical way.

Solving Systems of Equations by Graphing

Point verification in systems

• Substitute the given point's $x$ and $y$ values into each equation separately
  • If the point satisfies an equation, plugging in the values will make the left side equal the right side (equation balances)
  • Test the point in both equations to verify if it works for each one individually
• The point is a solution to the system if it satisfies both equations simultaneously
  • Must make both equations true at the same time to be a valid solution for the system (intersection point)
  • This solution is represented as an ordered pair (x, y)

Graphing for intersection points

• Graph each linear equation on the same coordinate plane
  • Choose two points for each equation by plugging in $x$ values and solving for $y$ (or vice versa)
  • Plot the two points for each equation and connect them with a straight line using a ruler
• The point(s) where the two lines intersect represent the solution(s) to the system
  • Lines will cross at the point(s) that satisfy both equations
  • Read the $x$ and $y$ coordinates of the intersection point(s) to find the solution(s)
  • Intersection point(s) give the values of the variables that make both equations true

Solution count from graph features

• One solution (consistent and independent)
  • The lines intersect at a single point (unique solution)
  • Most common case for two distinct linear equations
• No solution (inconsistent)
  • The lines are parallel and do not intersect at any point
  • Equations have same slope but different $y$-intercepts
• Infinitely many solutions (consistent and dependent)
  • The lines are coincident (overlapping) and intersect at every point
  • Equations are equivalent and have same slope and $y$-intercept

Real-world applications of graphing

• Identify the variables in the problem and assign them to $x$ and $y$
  • Variables often represent unknown quantities to solve for (price, time, distance)
• Write a system of equations based on the given information
  • Each equation should represent a relationship between the variables
  • Translate problem statements into linear equations (rate × time = distance)
• Graph the system of equations on the same coordinate plane
  • Find intersection point(s) and read coordinates to get variable values
• Interpret the solution(s) in the context of the real-world problem
  • The intersection point(s) provide the values that satisfy both conditions in the problem
  • Solutions give specific numbers for the unknown quantities (50 miles, $12 per hour)

Cartesian Coordinate System

• The coordinate plane is divided into four quadrants
• The origin (0, 0) is the point where the x-axis and y-axis intersect
• Points are plotted using ordered pairs (x, y) to represent their position relative to the origin