4.2 Graph Linear Equations in Two Variables

Linear equations are the building blocks of algebra, representing straight lines on a graph. They show how two variables relate, with each point on the line being a solution. Understanding these equations helps us model real-world relationships and solve problems visually.

Graphing linear equations involves plotting points and connecting them. We'll explore different types of lines, including vertical and horizontal, and learn about the coordinate plane. This knowledge forms the foundation for more complex algebraic concepts and problem-solving techniques.

Graphing Linear Equations

Solutions and graph representations

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  Graph linear functions | College Algebra View original

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• A linear equation in two variables represents a straight line when graphed on a coordinate plane
  • The line consists of all the points (ordered pairs) that satisfy the equation (solutions)
  • Each point on the line is a solution to the equation, meaning the x and y values make the equation true
• The graph provides a visual representation of the solution set, the collection of all solutions
  • Allows for a quick and intuitive understanding of the relationship between the variables
  • Shows the direction (increasing or decreasing) and steepness of the relationship

Graphing linear equations

• To graph a linear equation, plot points that satisfy the equation and connect them with a straight line
• Choose values for one variable (usually x) and solve for the corresponding values of the other variable (y)
  • Substitute the chosen x-value into the equation
  • Solve the resulting equation for y
• Plot the ordered pairs (x, y) on the coordinate plane
  • The x-coordinate represents the independent variable, the input value
  • The y-coordinate represents the dependent variable, the output value determined by the x-value and the equation
• Connect the plotted points with a straight line using a straightedge
• Only two points are needed to graph a line, but plotting more points confirms the line's accuracy

Vertical vs horizontal lines

• Vertical lines have equations in the form [x = a](https://www.fiveableKeyTerm:x_=_a), where $a$ is a constant
  • The x-coordinate is always the same, regardless of the y-coordinate
  • Vertical lines are perpendicular to the x-axis
  • To graph a vertical line, plot any two points with the given x-coordinate and connect them
  • Example: $x = 3$ is a vertical line that passes through (3, 0), (3, 1), (3, -2), and any other point with an x-coordinate of 3
• Horizontal lines have equations in the form [y = b](https://www.fiveableKeyTerm:y_=_b), where $b$ is a constant
  • The y-coordinate is always the same, regardless of the x-coordinate
  • Horizontal lines are perpendicular to the y-axis
  • To graph a horizontal line, plot any two points with the given y-coordinate and connect them
  • Example: $y = -1$ is a horizontal line that passes through (0, -1), (2, -1), (-3, -1), and any other point with a y-coordinate of -1

Coordinate Plane Components

• The coordinate plane is divided into four quadrants by the x-axis and y-axis
• The origin is the point where the x-axis and y-axis intersect, represented by the coordinates (0, 0)
• The x-axis is the horizontal line that runs through the origin
• The y-axis is the vertical line that runs through the origin
• An intercept is the point where a line crosses an axis (x-intercept for x-axis, y-intercept for y-axis)

Additional Concepts

• A function is a special type of relation where each input (x-value) corresponds to exactly one output (y-value)
  • Linear equations often represent functions, with each x-coordinate yielding a unique y-coordinate on the line