To determine the number of solutions for the given system of linear equations:
[tex]\[
\left\{
\begin{array}{c}
x + 7y = 35 \\
2x + y = 26
\end{array}
\right.
\][/tex]
let's carry out a step-by-step analysis.
### Step 1: Write the system of equations in standard form
The system is already in the standard form [tex]\(Ax + By = C\)[/tex], where:
1. The first equation is [tex]\(x + 7y = 35\)[/tex].
2. The second equation is [tex]\(2x + y = 26\)[/tex].
### Step 2: Identify coefficients
For the first equation ([tex]\(x + 7y = 35\)[/tex]):
- [tex]\(A1 = 1\)[/tex]
- [tex]\(B1 = 7\)[/tex]
- [tex]\(C1 = 35\)[/tex]
For the second equation ([tex]\(2x + y = 26\)[/tex]):
- [tex]\(A2 = 2\)[/tex]
- [tex]\(B2 = 1\)[/tex]
- [tex]\(C2 = 26\)[/tex]
### Step 3: Calculate the determinant of the coefficient matrix
The determinant [tex]\(\Delta\)[/tex] of the coefficient matrix is given by:
[tex]\[
\Delta = A1 \times B2 - A2 \times B1
\][/tex]
Substituting the values:
[tex]\[
\Delta = 1 \times 1 - 2 \times 7 = 1 - 14 = -13
\][/tex]
### Step 4: Analyze the determinant
- If the determinant ([tex]\(\Delta\)[/tex]) is non-zero ([tex]\(\Delta \neq 0\)[/tex]), the system has a unique solution.
- If the determinant ([tex]\(\Delta\)[/tex]) is zero ([tex]\(\Delta = 0\)[/tex]), further analysis is needed to determine if the system has infinitely many solutions or no solutions.
In this case, the determinant is [tex]\(\Delta = -13\)[/tex], which is non-zero.
### Conclusion
Since the determinant is non-zero ([tex]\(\Delta = -13\)[/tex]), the given system of equations has exactly one unique solution.
Thus, the number of solutions for the given system of equations is:
One.
