To solve the system of equations using cross multiplication, we need to express the given linear equations in the standard form [tex]\( a_1x + b_1y + c_1 = 0 \)[/tex] and [tex]\( a_2x + b_2y + c_2 = 0 \)[/tex]. Here are the steps we follow:
1. Write down the given equations:
The first equation is:
[tex]\[
1x + 1y + 1 = 0
\][/tex]
The second equation is:
[tex]\[
2x + 2y + 2 = 0
\][/tex]
2. Identify the coefficients:
- For the first equation [tex]\( a_1 = 1 \)[/tex], [tex]\( b_1 = 1 \)[/tex], and [tex]\( c_1 = 1 \)[/tex].
- For the second equation [tex]\( a_2 = 2 \)[/tex], [tex]\( b_2 = 2 \)[/tex], and [tex]\( c_2 = 2 \)[/tex].
3. Using cross multiplication:
The formulas to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using cross multiplication are as follows:
[tex]\[
x = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1}
\][/tex]
[tex]\[
y = \frac{c_1a_2 - c_2a_1}{a_1b_2 - a_2b_1}
\][/tex]
4. Calculate the denominator:
[tex]\[
\text{Denominator} = a_1b_2 - a_2b_1 = 1 \cdot 2 - 2 \cdot 1 = 2 - 2 = 0
\][/tex]
5. Evaluate the denominator:
The denominator is zero, which means that the determinant of the coefficient matrix is zero. This implies that the lines represented by the equations are either overlapping (infinitely many solutions) or parallel (no solution).
6. Conclusion:
Since the denominator is zero, the system of equations has either no solution or infinitely many solutions. In this particular case, both equations essentially represent the same line, which suggests that they have infinitely many solutions if all conditions for overlap are met exactly. If not, they would be parallel and hence have no solutions. Without more information to distinguish the exact scenario, we should conclude that:
[tex]\[
\text{The system of equations has either no solution or infinitely many solutions.}
\][/tex]
This completes our solution to the system of equations.
