Solve the following simultaneous equations:

a) [tex]\(x - 2y = 8\)[/tex]
b) [tex]\(3x + 2y = 2\)[/tex]
c) [tex]\(2x = 5\)[/tex]
d) [tex]\(y = x + 3\)[/tex]



Answer :

Certainly! Let's go through the system of equations step by step to determine if a solution exists:

1. We are given four equations:
[tex]\( x - 2y = 8 \)[/tex] (Equation 1)

[tex]\( 3x + 2y = 2 \)[/tex] (Equation 2)

[tex]\( 2x = 5 \)[/tex] (Equation 3)

[tex]\( y = x + 3 \)[/tex] (Equation 4)

2. Let's first solve Equation 3 for [tex]\( x \)[/tex]:
[tex]\[ 2x = 5 \][/tex]
[tex]\[ x = \frac{5}{2} = 2.5 \][/tex]

3. Substitute [tex]\( x = 2.5 \)[/tex] into Equation 4 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = x + 3 \][/tex]
[tex]\[ y = 2.5 + 3 = 5.5 \][/tex]

4. Now substitute [tex]\( x = 2.5 \)[/tex] and [tex]\( y = 5.5 \)[/tex] into Equations 1 and 2 to check for consistency.

Equation 1:
[tex]\[ x - 2y = 8 \][/tex]
Substituting [tex]\( x = 2.5 \)[/tex] and [tex]\( y = 5.5 \)[/tex]:
[tex]\[ 2.5 - 2(5.5) = 2.5 - 11 = -8.5 \][/tex]
[tex]\(-8.5 \ne 8\)[/tex], so this equation is not satisfied with these values.

Equation 2:
[tex]\[ 3x + 2y = 2 \][/tex]
Substituting [tex]\( x = 2.5 \)[/tex] and [tex]\( y = 5.5 \)[/tex]:
[tex]\[ 3(2.5) + 2(5.5) = 7.5 + 11 = 18.5 \][/tex]
[tex]\( 18.5 \ne 2 \)[/tex], so this equation is not satisfied either.

Since the values [tex]\( x = 2.5 \)[/tex] and [tex]\( y = 5.5 \)[/tex] do not satisfy Equations 1 and 2, we conclude that there is a contradiction. Therefore, the system of equations has no solution.

The solution to the given system of equations is:
[tex]\[ \boxed{\text{No solution}} \][/tex]

Answer:

For the simultaneous equations:

a) \( x - 2y = 8 \)

b) \( 3x + 2y = 2 \)

Add equations (a) and (b) to eliminate \( y \):

\[ 4x = 10 \]

\[ x = 2.5 \]

Substitute \( x = 2.5 \) into equation (a):

\[ 2.5 - 2y = 8 \]

\[ -2y = 5.5 \]

\[ y = -2.75 \]

So, \( x = 2.5 \) and \( y = -2.75 \).

For the equations:

c) \( 2x = 5 \)

d) \( y = x + 3 \)

From equation (c):

\[ x = 2.5 \]

Substitute \( x = 2.5 \) into equation (d):

\[ y = 2.5 + 3 \]

\[ y = 5.5 \]

So, \( x = 2.5 \) and \( y = 5.5 \).

Therefore, the solutions are:

1. For equations (a) and (b): \( (x, y) = (2.5, -2.75) \).

2. For equations (c) and (d): \( (x, y) = (2.5, 5.5) \).