A linear equation
represents a straight line on a graph. If two linear equations whose straight
lines intersect each other at a point, they are called simultaneous linear equations. The point of intersection is the common
solution to those linear equations, and hence the solution to the pair of
simultaneous equations.
Look at the example
of two linear equations x + y = 7 and x – y = 3 intersecting at a point (5, 2),
so they are simultaneous linear equations and the solution is (5, 2).
How to Solve Simultaneous Linear Equations Graphically?
To solve the simultaneous linear equations graphically, we find few pairs of solutions for each of the two given linear equations in two separate tables. The pair of solutions (points) of each equation are plotted in a graph and joined by a straight line. The coordinates of the point of intersection of these two straight lines are the solution of the given simultaneous equations.
Steps for solving simultaneous linear equations graphically:
Step 1: Take the first equation, and
equate it for one variable x or y in terms of other variable and find some
points (value of x and y) in a table.
Step 2: Take the second equation, and
equate it for one variable x or y in terms of other variable and find some
points (value of x and y) in a table.
Step 3: Plot the points of both
equations in a graph and join the lines.
Step 4: Find the point of the intersection of two straight lines (value of x and y) which is the solution of the given simultaneous equations.
This process
of solving simultaneous equations
graphically will be clear by the following worked-out examples.
Worked Out Examples
Example 1:
Solve x + y = 7 and x – y = 3 by graphical method.
Solution: Here,
x + y = 7 …………… (i)
x – y = 3 …………… (ii)
From equation (i),
x
+ y = 7
or, x
= 7 – y
x |
3 |
2 |
y |
4 |
5 |
∴ Points are: (3, 4) and (2, 5)
From equation (ii),
x – y = 3
or, x = 3 + y
x |
2 |
4 |
y |
-1 |
1 |
∴ Points are: (3, 4) and (2, 5)
Now, plotting the
points on the graph,
From the above graph,
point of intersection is (5, 2),
∴ Solution: x = 5 and y = 2
Example 2:
Solve 3x + y = 6 and x – 2y = 2 by graphical method.
Solution: Here,
3x + y = 6 …………… (i)
x – 2y = 2 …………… (ii)
From equation (i),
3x
+ y = 6
or, y
= 6 – 3x
x |
1 |
2 |
y |
3 |
0 |
∴ Points are: (1, 3) and (2, 0)
From equation (ii),
x – 2y = 2
or, x = 2 + 2y
x |
4 |
6 |
y |
1 |
2 |
∴ Points are: (4, 1) and (6, 2)
Now, plotting the
points on the graph,
From the graph,
point of intersection is (2, 0),
∴ Solution: x = 2 and y = 0
Example 3:
Solve 3x + 2y = 4 and 5x – y = 11 by graphical method.
Solution: Here,
3x + 2y = 4 …………… (i)
5x – y = 11 …………… (ii)
From equation (i),
3x + 2y = 4
or, 3x
= 4 – 2y
or, x
= (4 – 2y)/3
x |
0 |
2 |
y |
2 |
-1 |
∴ Points are: (0, 2) and (2, -1)
From equation (ii),
5x – y = 11
or, 5x – 11 = y
or, y = 5x – 11
x |
2 |
3 |
y |
-1 |
4 |
∴ Points are: (2, -1) and (3, 4)
Now, plotting the
points on the graph,
From the graph, point of intersection is (2, -1),
∴ Solution: x = 2 and y = -1
Do you have any questions regarding the Simultaneous Equations Graphically?
You can ask your questions or problems in the comment section below.
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