Linear Equations and Graphing

Equation Analysis

• The given equation is a linear equation in two variables, x and y.
• It represents a straight line on a Cartesian coordinate system.
• The equation can be written in the form (y = 2x - 4). This is the slope-intercept form, where the slope is 2 and the y-intercept is -4.

Graphing the Equation

• To graph the line, start by plotting the y-intercept at (0, -4).
• Use the slope (2, which can be expressed as rise/run = 2/1) to find another point. From the y-intercept, move one unit to the right (run=1), and two units up (rise=2). This gives you the point (1, -2).
• To verify, substitute x = 1 and y = -2 into the original equation, which yields 2(1) - (-2) = 4, validating the point.
• Continue this process to plot more points and sketch the line.

Solving for x and y

• Given a value for x, you can solve for y in the equation y = 2x - 4.
• Given a value for y, you can solve for x in the equation x = (y+4)/2.
• There are infinitely many solutions to the equation as there are infinite points on the line, each corresponding to a unique value of x and y.
• Example solutions:
  • If x = 2, then y = 2(2) - 4 = 0, giving the point (2, 0).
  • If x = 0, then y = 2(0) - 4 = -4, giving the point (0, -4).
  • If y = 2, then 2 = 2x - 4, so 2x = 6, and x = 3, giving the point (3, 2).

Slope-Intercept Form

• The equation can be expressed in slope-intercept form: (y = mx + b), where:
  • m is the slope of the line, representing the rate of change of y with respect to x.
  • b is the y-intercept, the point where the line intersects the y-axis.

Standard Form

• The equation can also be written in standard form: (Ax + By = C), where A, B, and C are integers. In this case, (2x - y = 4) is already in standard form.

Solutions and Points

• Every point on the line determined by the equation 2x - y = 4 represents a solution to the equation.

Applications

• The equation 2x - y = 4 has applications in various fields:
  • Physics: Modelling linear relationships between quantities.
  • Economics: Representing supply and demand.
  • Engineering: Describing linear patterns.
  • Mathematics: General use.

Important Considerations

• The equation is linear, resulting in a straight-line graph.
• It defines a single straight line with an infinite number of points satisfying the equation.
• The equation shows a relationship between x and y, displaying how a change in x affects y and vice versa.