Question

line e: y = - 1/3x+6

line f: y = 4x-7 Which statements about the system are true? (Choose two.)

Responses
The solution to the system is (3,5) It is impossible for y to equal both 4x-7 and -1/3x +6 simultaneously; therefore, the system has no solution.
It is impossible for y to equal both 4 x minus 7 and negative 1 third x plus 6 simultaneously; therefore, the system has no solution.
The system is consistent because the lines are perpendicular.
The system is consistent because the lines are NOT parallel.
The system is inconsistent because the lines e and f intersect

Answer 1

To analyze the two lines given by the equations:

1. Line e: \( y = -\frac{1}{3}x + 6 \)
2. Line f: \( y = 4x - 7 \)

### Step 1: Determine the slopes
- For line e, the slope is \( -\frac{1}{3} \).
- For line f, the slope is \( 4 \).

### Step 2: Identify if the lines are parallel or perpendicular
- The lines are not parallel since they have different slopes.
- The lines are not perpendicular either, as the product of their slopes \( (-\frac{1}{3}) \times 4 = -\frac{4}{3} \) is not equal to -1.

### Step 3: Check for an intersection
- Since the lines are not parallel, they will intersect at one point.

### Step 4: Find the intersection point
To find the intersection point, set the equations equal:
\[
-\frac{1}{3}x + 6 = 4x - 7
\]

Multiply through by 3 to eliminate the fraction:
\[
-x + 18 = 12x - 21
\]

Combine like terms:
\[
18 + 21 = 12x + x
\]
\[
39 = 13x
\]
\[
x = 3
\]

Now substitute \( x = 3 \) back into either original equation to find \( y \):
Using line f:
\[
y = 4(3) - 7 = 12 - 7 = 5
\]

Thus, the intersection point (solution to the system) is \( (3, 5) \).

### Conclusions
Now we can evaluate the statements:
1. **The solution to the system is (3,5)** - **True**
2. **It is impossible for \( y \) to equal both \( 4x - 7 \) and \( -\frac{1}{3}x + 6 \) simultaneously; therefore, the system has no solution.** - **False**
3. **The system is consistent because the lines are perpendicular.** - **False**
4. **The system is consistent because the lines are NOT parallel.** - **True**
5. **The system is inconsistent because the lines e and f intersect.** - **False**

### Final Answers
The two true statements are:
- The solution to the system is (3,5).
- The system is consistent because the lines are NOT parallel.