To analyze the system of two linear equations, we'll start with the equations given:
- \( x - 5y = 7 \)
- \( 3x - 2y = 4 \)
Step 1: Graphing the Equations
To graph these equations, we can rewrite each equation in slope-intercept form \( y = mx + b \).
For the first equation: \( x - 5y = 7 \)
We can isolate \( y \):
\[ -5y = -x + 7 \]
\[ y = \frac{1}{5}x - \frac{7}{5} \]
This has a slope \( m = \frac{1}{5} \) and y-intercept \( b = -\frac{7}{5} \).
For the second equation: \( 3x - 2y = 4 \)
Again, isolating \( y \):
\[ -2y = -3x + 4 \]
\[ y = \frac{3}{2}x - 2 \]
This has a slope \( m = \frac{3}{2} \) and y-intercept \( b = -2 \).
Step 2: Finding the Intersection (Solution)
To find the solution of the system (the point where the two lines intersect), we can solve the equations simultaneously. We can substitute one equation into the other or use elimination.
Substitution Method
From the first equation \( x = 5y + 7 \). Now, substitute this into the second equation:
\[ 3(5y + 7) - 2y = 4 \]
Expanding this gives:
\[ 15y + 21 - 2y = 4 \]
Combining like terms:
\[ 13y + 21 = 4 \]
Now, isolate \( y \):
\[ 13y = 4 - 21 \]
\[ 13y = -17 \]
\[ y = -\frac{17}{13} \]
Substituting back to find \( x \)
Now substitute \( y = -\frac{17}{13} \) back into the equation \( x = 5y + 7 \):
\[ x = 5\left(-\frac{17}{13}\right) + 7 \]
Calculating this gives:
\[ x = -\frac{85}{13} + 7 \]
Converting \( 7 \) to a fraction:
\[ x = -\frac{85}{13} + \frac{91}{13} \]
\[ x = \frac{6}{13} \]
Conclusion
The solution to the system of equations is:
\[ \left( \frac{6}{13}, -\frac{17}{13} \right) \]
Graphical Interpretation: The graphs of these two equations will intersect at the point \(\left( \frac{6}{13}, -\frac{17}{13} \right)\), which represents the solution common to both equations.