Answer :
To determine the number of solutions for the given system of equations, we need to closely analyze the given set of equations:
[tex]\[ \begin{array}{l} 2x + 4y = 15 \\ 2x + 4y = 13 \end{array} \][/tex]
Let's follow these steps:
1. Compare the two equations:
The left-hand sides of both equations are identical:
[tex]\[ 2x + 4y \][/tex]
This means that for the two equations to have any solutions, the right-hand sides must be the same as well. However, the right-hand sides are different:
[tex]\[ 15 \quad \text{and} \quad 13 \][/tex]
So we have:
[tex]\[ 2x + 4y = 15 \][/tex]
[tex]\[ 2x + 4y = 13 \][/tex]
2. Subtract one equation from the other:
Subtract the second equation from the first:
[tex]\[ (2x + 4y) - (2x + 4y) = 15 - 13 \][/tex]
Simplifying this, we get:
[tex]\[ 0 = 2 \][/tex]
This resulting equation, \(0 = 2\), is a contradiction.
3. Conclusion:
A contradiction indicates that the system of equations is inconsistent, meaning there are no values of \(x\) and \(y\) that satisfy both equations simultaneously. Therefore, the system of equations has no solutions.
Thus, the system of equations:
[tex]\[ \begin{array}{l} 2x + 4y = 15 \\ 2x + 4y = 13 \end{array} \][/tex]
has [tex]\(0\)[/tex] solutions.
[tex]\[ \begin{array}{l} 2x + 4y = 15 \\ 2x + 4y = 13 \end{array} \][/tex]
Let's follow these steps:
1. Compare the two equations:
The left-hand sides of both equations are identical:
[tex]\[ 2x + 4y \][/tex]
This means that for the two equations to have any solutions, the right-hand sides must be the same as well. However, the right-hand sides are different:
[tex]\[ 15 \quad \text{and} \quad 13 \][/tex]
So we have:
[tex]\[ 2x + 4y = 15 \][/tex]
[tex]\[ 2x + 4y = 13 \][/tex]
2. Subtract one equation from the other:
Subtract the second equation from the first:
[tex]\[ (2x + 4y) - (2x + 4y) = 15 - 13 \][/tex]
Simplifying this, we get:
[tex]\[ 0 = 2 \][/tex]
This resulting equation, \(0 = 2\), is a contradiction.
3. Conclusion:
A contradiction indicates that the system of equations is inconsistent, meaning there are no values of \(x\) and \(y\) that satisfy both equations simultaneously. Therefore, the system of equations has no solutions.
Thus, the system of equations:
[tex]\[ \begin{array}{l} 2x + 4y = 15 \\ 2x + 4y = 13 \end{array} \][/tex]
has [tex]\(0\)[/tex] solutions.
