{"id":9964,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9964"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"solving-using-graphical-methods","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/solving-using-graphical-methods\/","title":{"rendered":"Solving Using Graphical Methods"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Linear Equation in two variables<\/strong><\/h2>\n

Chapter: <\/strong>Solving Using Graphical Method<\/strong><\/h3>\n

Reference: – Definition of Linear Equations in Two Variables, Standard Form of Linear Equations, Cartesian Coordinate Plane, Plotting Points on the Cartesian Plane, Graph of a Linear Equation, Finding X-intercept and Y-intercept, Slope of a Line, Slope-Intercept Form, Point-Slope Form, Graphing Linear Equations Using a Table of Values, solving a System of Linear Equations Graphically, Types of Solutions for Linear Systems, Checking Solutions by Substitution<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • Definition of Linear Equations in Two Variables & Standard Form of Linear Equations<\/li>\n
  • Graph of a Linear Equation & Finding X-intercept and Y-intercept<\/li>\n
  • Graphing Linear Equations Using a Table of Values & solving a System of Linear Equations Graphically<\/li>\n
  • Types of Solutions for Linear Systems, Checking Solutions by Substitution<\/li>\n<\/ul>\n

    Definition of Linear Equations in Two Variables:<\/strong><\/p>\n

    A linear equation in two variables represents a relationship where each variable appears with an exponent of one, and the graph of the relationship forms a straight line when plotted on a coordinate plane.<\/p>\n

    Standard Form of Linear Equations:<\/strong><\/p>\n

    This is the conventional way of writing linear equations where all variable terms and constants are placed on one side, typically written as<\/p>\n

    Ax+By=C, making it easier to analyse and compare.<\/p>\n

    Cartesian Coordinate Plane:<\/strong><\/p>\n

    A two-dimensional grid defined by perpendicular horizontal and vertical axes (x-axis and y-axis), used to locate points, lines, and graphs representing algebraic relationships.<\/p>\n

    Plotting Points on the Cartesian Plane:<\/strong><\/p>\n

    The process of marking locations on the coordinate plane using ordered pairs (x,y), where x represents the horizontal position and y the vertical position.<\/p>\n

    Graph of a Linear Equation:<\/strong><\/p>\n

    A visual representation of all solutions to a linear equation, depicted as a straight line that connects multiple solution points on the coordinate plane.<\/p>\n

    Finding X-intercept and Y-intercept:<\/strong><\/p>\n

    Identifying the points where the graph of the equation crosses the x-axis and y-axis, which helps in quickly sketching the graph of the line.<\/p>\n

    Slope of a Line:<\/strong><\/p>\n

    A measure of how steep a line is on the graph, representing the rate at which the dependent variable changes with respect to the independent variable.<\/p>\n

    Slope-Intercept Form:<\/strong><\/p>\n

    A way to express linear equations that highlights both the slope and the y-intercept, making it easier to graph the line directly by identifying these two components.<\/p>\n

    Point-Slope Form:<\/strong><\/p>\n

    An alternative form of linear equations that emphasizes a specific point on the line and the slope, useful for quickly writing equations when slope and a point are known.<\/p>\n

    Graphing Linear Equations Using a Table of Values:<\/strong><\/p>\n

    A method where several input values (x-values) are chosen, and corresponding output values (y-values) are calculated, creating points that can be plotted to draw the line.<\/p>\n

    Solving a System of Linear Equations Graphically:<\/strong><\/p>\n

    A technique where two linear equations are plotted on the same graph to visually identify their point of intersection, representing the simultaneous solution to both equations.<\/p>\n

    Types of Solutions for Linear Systems:<\/strong><\/p>\n

    A classification of the possible outcomes for systems of linear equations: having exactly one solution (lines intersect), no solution (parallel lines), or infinitely many solutions (overlapping lines).<\/p>\n

    Checking Solutions by Substitution:<\/strong><\/p>\n

    After graphing, verifying the accuracy of the intersection point by substituting the x and y values back into both original equations to confirm they satisfy both.<\/p>\n

    Graphical Interpretation of Consistent and Inconsistent Systems:<\/strong><\/p>\n

    Understanding how the visual layout of lines on the graph (whether they intersect or not) reflects whether the system has at least one solution (consistent) or none (inconsistent).<\/p>\n

    Real-World Applications of Graphical Solutions:<\/strong><\/p>\n

    Applying the graphical method to practical scenarios such as budgeting, speed and distance problems, and resource management, where two quantities are linearly related.<\/p>\n

    \u00a0<\/p>\n

    Example: –<\/u><\/strong><\/p>\n

    Two delivery companies offer different pricing models for shipping a package:<\/p>\n

      \n
    • Company A<\/strong> charges a fixed fee of $10<\/strong> plus $2 per kilometre<\/strong> travelled.<\/li>\n
    • Company B<\/strong> charges a fixed fee of $5<\/strong> plus $3 per kilometre<\/strong> travelled.<\/li>\n
    • By graphing both cost models on the same coordinate plane, find the distance at which both companies charge the same total price.<\/li>\n
    • Also, state which company is cheaper before and after this distance.<\/li>\n<\/ul>\n

      Solution: –<\/u><\/strong>
      \n\u00a0<\/p>\n

      Step 1: Formulating Linear Equations<\/strong><\/p>\n

      Let x<\/strong> represent the distance in kilometres.
      \nLet y<\/strong> represent the total cost in dollars.<\/p>\n

      \"\"<\/p>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Step 2: Set up Table of Values (for Graphing)<\/p>\n

      \"\"<\/p>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Step 3: Graphing<\/strong><\/p>\n

      On graph paper or coordinate plane:<\/p>\n

      \"\"<\/p>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Step 4: Intersection Point (Solution)<\/strong><\/p>\n

      From both the table and graph:<\/p>\n

        \n
      • The lines intersect at point (5, 20)<\/strong>.<\/li>\n<\/ul>\n

        This means:
        \nAt 5 kilometres<\/strong>, both companies charge exactly $20<\/strong>.<\/p>\n

        Step 5: Analysis<\/strong><\/p>\n

          \n
        • For distances less than 5 km<\/strong>, Company B is cheaper<\/strong>.<\/li>\n
        • For distances more than 5 km<\/strong>, Company A becomes cheaper<\/strong>.<\/li>\n<\/ul>\n

          Here are five conclusive points: –<\/u><\/strong><\/p>\n

          1. Visual Representation of Solutions:<\/strong><\/p>\n

          Graphing provides a clear, visual method to find solutions to linear equations and systems of equations by identifying where lines intersect on the coordinate plane.<\/p>\n

          \u00a0<\/p>\n

          2. Understanding Relationship Between Variables:<\/strong><\/p>\n

          By observing the slope and intercepts, students develop an intuitive understanding of how changes in one variable affect the other in a linear relationship.<\/p>\n

          \u00a0<\/p>\n

          3. Comparison Between Systems:<\/strong><\/p>\n

          The graphical method helps distinguish between systems that have one solution (intersecting lines), no solution (parallel lines), and infinitely many solutions (coinciding lines).<\/p>\n

          \u00a0<\/p>\n

          4. Real-World Problem Solving:<\/strong><\/p>\n

          Graphing is a practical tool for solving real-world problems where two quantities change at a constant rate and their interaction needs to be analysed visually.<\/p>\n

          \u00a0<\/p>\n

          5. Reinforcement of Algebraic Concepts:<\/strong><\/p>\n

          Graphing linear equations strengthens understanding of slope, intercepts, and solution verification, reinforcing connections between algebraic and geometric representations.<\/p>\n

          \u00a0<\/p>\n

          \u00a0<\/p>\n

          <\/p>\n