Solving Simultaneous Equation Graphically

Solving System ofSimultaneous Equations Graphically

Lesson Objectives

Understand the graphical method of solving systems of linear equations.
Learn how to plot two equations on the same coordinate plane.
Determine the solution of simultaneous equations from their graphs.
Interpret the meaning of the point(s) of intersection graphically.

Lesson Introduction

This lesson introduces the method of solving systems of linear equations by graphing. You will learn how to plot two equations on the same axes and find their point of intersection, which represents the solution to the system. This graphical approach helps visualize the solution and understand the relationship between equations.

Lesson Content

Graphical Method of Solving Systems of Equations

The graphical method involves plotting each equation on the coordinate plane and identifying the point(s) where the graphs intersect. For two linear equations:

$\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$

Plot both lines by finding intercepts or points, then draw them on the same graph. The coordinates of their intersection point satisfy both equations simultaneously.

Steps to Solve by Graphical Method

Rewrite each equation in slope-intercept form: $y = mx + c$ .
Plot the graph of each equation on the same coordinate axes.
Identify the point where the two graphs intersect.
The coordinates of the intersection point ( $(x, y)$ ) are the solution to the system.

Examples

Example 1:
Solve graphically the system:

\begin{cases} x + 2y = 10 \\ x - y = 1 \end{cases}

Solution:
Step 1: Rewrite in slope-intercept form:

\begin{cases} 2y = 10 - x \Rightarrow y = \frac{10 - x}{2} \\ x - y = 1 \Rightarrow y = x - 1 \end{cases}

Step 2: Plot both lines on the same graph.
Step 3: The point of intersection is where:

y = \frac{10 - x}{2} = x - 1

Step 4: Solve for

x

\frac{10 - x}{2} = x - 1 \Rightarrow 10 - x = 2x - 2 \Rightarrow 10 + 2 = 2x + x \Rightarrow 12 = 3x \Rightarrow x = 4

Step 5: Find

y

y = x - 1 = 4 - 1 = 3

Therefore, the solution is

(4, 3)

Example 2:
Solve graphically:

\begin{cases} y = 3x - 1 \\ y = -2x + 4 \end{cases}

Solution:
Step 1: Plot both lines.
Step 2: Find the point of intersection by setting:

3x - 1 = -2x + 4

Step 3: Solve for

x

3x + 2x = 4 + 1 \Rightarrow 5x = 5 \Rightarrow x = 1

Step 4: Find

y

y = 3(1) - 1 = 2

Solution is

(1, 2)

Example 3:
Solve graphically:

\begin{cases} y = 2x^2 \\ y = 4x - 2 \end{cases}

Solution:
Plot both curves and find intersection points by solving:

2x^2 = 4x - 2 \Rightarrow 2x^2 - 4x + 2 = 0

Divide both sides by 2:

x^2 - 2x + 1 = 0

Factorize:

(x - 1)^2 = 0 \Rightarrow x = 1

Substitute back:

y = 2(1)^2 = 2

Solution is

(1, 2)

Interactive Graph

Exercises

[WAEC] Solve graphically: $x + 2y = 10$ , $x - y = 1$ [Past Question]
Solve: $y = 3x - 1$ , $y = -2x + 4$
Solve: $y = 2x^2$ , $y = 4x - 2$
[WASSCE] Solve graphically: $x + y = 6$ , $2x - y = 3$ [Past Question]
Solve: $y = 4x - 5$ , $y = -x + 7$
[NECO] Solve: $y = x^2 + 2x$ , $y = 3x + 1$ [Past Question]
Solve: $x - y = 2$ , $2x + y = 8$
Solve: $y = 3x^2 - 1$ , $y = x + 4$
[WAEC] Solve graphically: $y = x^2$ , $y = 5 - x$ [Past Question]
Solve: $y = 2x - 3$ , $y = x^2 - 4$

Conclusion/Recap

This lesson has covered solving systems of linear and quadratic equations by graphical method. You should now be able to plot equations, find their points of intersection, and interpret these solutions graphically to solve simultaneous equations.