To find the solution to the system of linear equations:
[tex]\[
\begin{cases}
3x + 5y = 14 \\
6x - 4y = 9
\end{cases}
\][/tex]
we can use various methods (algebraic, graphical, or computational). For this problem, we will estimate the solution and then pick the correct option.
### Step-by-Step Solution:
1. Equation Analysis:
- The first equation is [tex]\(3x + 5y = 14\)[/tex].
- The second equation is [tex]\(6x - 4y = 9\)[/tex].
2. Determine Intersection:
- When solving systems of linear equations graphically, we look for the point where the two lines intersect. This point is the solution [tex]\((x, y)\)[/tex] to both equations.
3. Graphical Approach:
- Graph the first equation [tex]\(3x + 5y = 14\)[/tex]:
To find the intercepts:
- Set [tex]\(x=0\)[/tex]: [tex]\(3(0) + 5y = 14 \Rightarrow y = \frac{14}{5} = 2.8\)[/tex]
- Set [tex]\(y=0\)[/tex]: [tex]\(3x + 5(0) = 14 \Rightarrow x = \frac{14}{3} \approx 4.67\)[/tex]
The line passes through [tex]\((0, 2.8)\)[/tex] and [tex]\((4.67, 0)\)[/tex].
- Graph the second equation [tex]\(6x - 4y = 9\)[/tex]:
To find the intercepts:
- Set [tex]\(x=0\)[/tex]: [tex]\(6(0) - 4y = 9 \Rightarrow -4y = 9 \Rightarrow y = -\frac{9}{4} = -2.25\)[/tex]
- Set [tex]\(y=0\)[/tex]: [tex]\(6x - 4(0) = 9 \Rightarrow x = \frac{9}{6} = 1.5\)[/tex]
The line passes through [tex]\((0, -2.25)\)[/tex] and [tex]\((1.5, 0)\)[/tex].
4. Identifying the Correct Option:
By solving the equations, we find the intersection point of these two lines. Comparing the solutions with the given options:
- We observe the options:
[tex]\[
\begin{aligned}
&\text{A.} \left(\frac{7}{3}, -\frac{7}{2}\right) \approx (2.33, -3.5) \\
&\text{B.} \left(-\frac{5}{2}, -\frac{7}{2}\right) \approx (-2.5, -3.5) \\
&\text{C.} \left(\frac{5}{2}, \frac{4}{3}\right) \approx (2.5, 1.33) \\
&\text{D.} \left(\frac{4}{3}, \frac{5}{2}\right) \approx (1.33, 2.5) \\
\end{aligned}
\][/tex]
Given the calculated solution [tex]\((2.4047619047619047, 1.3571428571428572)\)[/tex], we see that this closely matches with option C: [tex]\(\left(\frac{5}{2}, \frac{4}{3}\right)\)[/tex].
### Conclusion:
Therefore, the correct answer is:
C. [tex]\(\left(\frac{5}{2}, \frac{4}{3}\right)\)[/tex]
