3.3 Simultaneous Equations on Graphs
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Learning Objectives
- Understand the graphical interpretation of simultaneous equations
- Solve simultaneous linear equations graphically
- Interpret the graphical solutions for linear-quadratic systems
- Identify cases with no solutions, one solution, or multiple solutions
- Apply graphical methods to real-world problems involving simultaneous equations
Key Concepts
Introduction to Simultaneous Equations on Graphs
Simultaneous equations can be solved graphically by plotting their equations and finding points of intersection.
For two linear equations, the solution is where the two straight lines meet.
For a linear and a quadratic equation, the solutions are where the straight line intersects the curve.
The number of solutions corresponds to the number of intersection points between the graphs.
Solving Two Linear Equations Graphically
To solve two linear equations graphically, plot both equations on the same set of axes.
The point(s) where the lines intersect represent the solution(s).
If the lines are parallel, there are no solutions (inconsistent system).
If the lines coincide (identical equations), there are infinitely many solutions (dependent system).
Example 1
Solve graphically: y = 2x + 1 y = -x + 4
Solution:
Step 1: Plot y = 2x + 1 (a straight line with gradient 2 and y-intercept 1). Step 2: Plot y = -x + 4 (a straight line with gradient -1 and y-intercept 4). Step 3: Identify the intersection point, which is (1, 3). Thus, the solution is x = 1, y = 3.
Example 2
Solve graphically: 3x + 2y = 6 6x + 4y = 12
Solution:
Step 1: Rearrange both equations into y = mx + c form. Equation 1: y = -3/2x + 3 Equation 2: y = -3/2x + 3 (same equation as the first). Step 2: Since both equations describe the same line, they coincide. Thus, there are infinitely many solutions.
Solving a Linear and a Quadratic Equation Graphically
To solve a system with one quadratic and one linear equation graphically, plot both on the same set of axes.
The quadratic equation represents a parabola, and the linear equation represents a straight line.
The number of solutions depends on the number of intersection points:
- No intersection: No real solutions.
- One intersection: One repeated solution (the line is tangent to the parabola).
- Two intersections: Two distinct solutions.
Example 1
Solve graphically: y = x² - 4 y = x
Solution:
Step 1: Plot y = x² - 4 (a parabola with vertex at (0, -4)). Step 2: Plot y = x (a straight line passing through the origin with a gradient of 1). Step 3: Identify the intersection points. By solving algebraically: x = x² - 4 x² - x - 4 = 0 (x - 2)(x + 2) = 0 x = 2 or x = -2 Step 4: Find corresponding y-values. For x = 2, y = 2. For x = -2, y = -2. Thus, the solutions are (2,2) and (-2,-2).
Example 2
Solve graphically: y = x² - 3x + 2 y = 2
Solution:
Step 1: Plot y = x² - 3x + 2 (a parabola with roots at x = 1 and x = 2). Step 2: Plot y = 2 (a horizontal line at y = 2). Step 3: Identify the intersection points. By solving algebraically: 2 = x² - 3x + 2 x² - 3x = 0 x(x - 3) = 0 x = 0 or x = 3 Step 4: Find corresponding y-values. For x = 0, y = 2. For x = 3, y = 2. Thus, the solutions are (0,2) and (3,2).
Special Cases and Considerations
If a linear equation is a tangent to a quadratic graph, there is exactly one solution (repeated root).
If the quadratic and the linear equation do not intersect, there are no real solutions.
Graphical methods can provide approximate solutions, but algebraic methods should be used for exact answers.
Example 1
Determine the number of solutions for: y = x² - 2x + 3 y = x + 4
Solution:
Step 1: Find points of intersection by solving algebraically. x + 4 = x² - 2x + 3 x² - 3x - 1 = 0 Step 2: Find the discriminant of x² - 3x - 1 = 0. Δ = (-3)² - 4(1)(-1) = 9 + 4 = 13 (positive, so two solutions). Step 3: Conclude that the graphs intersect at two points.
Example 2
Determine whether y = x² + 2x + 5 and y = x - 1 have any solutions.
Solution:
Step 1: Solve algebraically. x - 1 = x² + 2x + 5 x² + x + 6 = 0 Step 2: Find the discriminant. Δ = (1)² - 4(1)(6) = 1 - 24 = -23 (negative, so no real solutions). Step 3: Conclude that the graphs do not intersect.
Applications of Simultaneous Equations on Graphs
Graphical solutions to simultaneous equations are useful in physics, economics, and engineering.
Examples include equilibrium points in supply and demand models, projectile motion, and intersection of cost and revenue functions.
Example 1
A companys cost and revenue functions are given by: Cost: C = 5x + 20 Revenue: R = -x² + 12x Find the break-even points where cost equals revenue.
Solution:
Step 1: Set C = R. 5x + 20 = -x² + 12x x² - 7x - 20 = 0 Step 2: Solve for x. (x - 5)(x - 4) = 0 x = 5 or x = 4 Thus, the break-even points occur at x = 4 and x = 5.
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