The graphical method is a technique also referred to as the geometric method used to solve linear equations systems. This method involves developing equations based on constraints and an objective function. To obtain solutions, the graphical method requires several steps. However, in this article, we will focus on explaining in detail how to solve linear equations systems using the first algebraic method, known as the Substitution Method.
Let’s go through the example,
Y = 2x + 4, 3x + y = 9
Now we can substitute y in the second equation with the first equation since we can write the second equation in terms of y(y=y) as
3x + y = 9
y=2x + 4
3x + (2x + 4) =9
5x + 4 =9
5x = 9  4
5x = 5
x = 1
This value of x that is equal to 1 can then be used to find the value of y by substituting 1 with x example in the first equation,
Y = 2x + 4y = 2x + 4 = 2 x 1 + 4y
= 2 x 1 + 4 = 6
Y = 6
Therefore, the value of y is equal to 6.
The solution of the linear system is equal to (1, 6).
We can use the substitution method even if both equations of the linear system are in standard form. We can just begin by solving one of the equations for one of its variables.
To solve for the values of x and y, we can use the substitution method:
From the first equation, we have:
x + y = 5
y = 5  x (subtracting x from both sides)
We can substitute this value of y into the second equation:
3x + y = 11
3x + (5  x) = 11 (substitute y = 5  x)
2x + 5 = 11
2x = 11  5
2x = 6
x = 3
Now that we know the value of x, we can substitute it into one of the original equations to solve for y. Let's use the first equation:
x + y = 5
3 + y = 5 (substitute x = 3)
y = 2
Therefore, the solution for the values of x and y is:
x = 3, y = 2.
Question 2: Solve for the values of ‘x’ and ‘y’: 2x + 6y = 10, 1x  2y = 15
To solve for the values of 'x' and 'y' in the equations:
2x + 6y = 10
1x  2y = 15
We can use the substitution method or the elimination method. Here's how to solve it using the substitution method:
Solve one of the equations for one variable (in terms of the other variable):
Let's solve the second equation for 'x':
1x  2y = 15
1x = 2y + 15
Substitute the expression found in step 1 into the other equation:
2(2y + 15) + 6y = 10
4y + 30 + 6y = 10
10y = 20
y = 2
Substitute the value of 'y' back into one of the original equations to solve for 'x':
2x + 6y = 10
2x + 6(2) = 10
2x  12 = 10
2x = 22
x = 11
Therefore, the solution for the system of equations is x = 11 and y = 2.
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