To determine which statement is true about the given system of equations:
1. [tex]\(-3x + 4y = 12\)[/tex]
2. [tex]\(\frac{1}{4}x - \frac{1}{3}y = 1\)[/tex]
we can solve the system of equations step by step.
### Step 1: Rewrite the equations in a simpler form
The second equation can be rewritten by eliminating the fractions. Multiply every term by 12 (the least common multiple of 4 and 3) to clear fractions:
[tex]\[
12 \left(\frac{1}{4}x - \frac{1}{3}y = 1\right)
\][/tex]
[tex]\[
3x - 4y = 12
\][/tex]
Now the system of equations looks like this:
1. [tex]\(-3x + 4y = 12\)[/tex]
2. [tex]\(3x - 4y = 12\)[/tex]
### Step 2: Add the equations
Add the two equations together to see if we can eliminate one of the variables:
[tex]\[
(-3x + 4y) + (3x - 4y) = 12 + 12
\][/tex]
Simplifying the left-hand side:
[tex]\[
(-3x + 3x) + (4y - 4y) = 24
\][/tex]
[tex]\[
0 + 0 = 24
\][/tex]
This results in:
[tex]\[
0 = 24
\][/tex]
This is a contradiction, which means the two equations represent parallel lines that never intersect. Hence, there is no solution to this system of equations.
### Conclusion
The correct statement is:
- The system of equations has no solution; the two lines are parallel.
