Understanding Linear Graphs and Equations

• Linear graphs show relationships where the change between plotted values remains constant.
• Simultaneous linear equations involve multiple linear equations sharing the same variables; the solution occurs where the lines intersect, overlap, or do not intersect.

Linear Graphs

• They visually represent linear equations.
• Linear equations follow the format y = mx + c:
  • x and y are variables.
  • m represents the slope or gradient.
  • c is the y-intercept.
• The slope (m) indicates the line's steepness and direction:
  • A positive slope means the line goes upwards from left to right.
  • A negative slope means the line goes downwards from left to right.
  • A zero slope results in a horizontal line.
• The y-intercept (c) is the value of y when x = 0.
• To plot:
  • Select several x values.
  • Calculate the corresponding y values.
  • Plot (x, y) coordinates.
  • Draw a straight line.
• Two points are sufficient, but a third can verify accuracy.
• The x-intercept is where the line crosses the x-axis (y = 0).
• Linear graphs show the relationship between two variables.
• Gradients are calculated as rise/run (vertical change divided by horizontal change).

Simultaneous Linear Equations

• These are sets of two or more linear equations containing the same variables.
• Solutions satisfy all equations simultaneously.
• Graphically, the solution is the intersection point of the lines.
• Three outcomes exist when solving two simultaneous linear equations:
  • Unique Solution: Lines intersect at one point, yielding a unique x and y value.
  • No Solution: Parallel lines that do not intersect indicate no solution exists.
  • Infinite Solutions: Identical lines mean infinite solutions; any point on the line satisfies both equations.

Methods for Solving Simultaneous Linear Equations

• Graphical Method:
  • Plot both equations on the same graph.
  • Find the intersection point.
  • Best for visualization but less accurate for non-integer solutions.
• Substitution Method:
  • Solve one equation for one variable.
  • Substitute into the other equation.
  • Solve for the single variable.
  • Substitute the value back to find the other variable.
• Elimination Method (Addition or Subtraction Method):
  • Multiply equations to match or oppose coefficients of one variable.
  • Add or subtract equations to eliminate a variable.
  • Solve for the remaining variable.
  • Substitute back to find the eliminated variable’s value.

Applications of Linear Graphs and Simultaneous Equations

• Solving linear-relationship word problems.
• Finding break-even points.
• Determining optimal solutions in engineering and economics.
• Modeling supply and demand curves.
• Calculating mixtures and concentrations.
• Solving systems of forces in physics.
• Predicting trends from linear data.

Example of Solving Simultaneous Equations Using Substitution

• Equations:
  • y = 2x + 1
  • x + y = 4
• Substituting the first equation into the second:
  • x + (2x + 1) = 4
  • 3x + 1 = 4
  • 3x = 3
  • x = 1
• Substituting x = 1 back into y = 2x + 1:
  • y = 2(1) + 1
  • y = 3
• Solution: x = 1 and y = 3.

Example of Solving Simultaneous Equations Using Elimination

• Equations:
  • 2x + y = 7
  • x - y = 2
• Adding the equations to eliminate y:
  • (2x + y) + (x - y) = 7 + 2
  • 3x = 9
  • x = 3
• Substituting x = 3 back into x - y = 2:
  • 3 - y = 2
  • -y = -1
  • y = 1
• Solution: x = 3 and y = 1.

Explore linear graphs and simultaneous linear equations, focusing on identifying slopes, intercepts, and solutions. Learn to plot linear graphs using the equation y = mx + c, and understand how slopes affect the direction of the line. Discover how to solve simultaneous equations by finding intersection points.