Question

On a coordinate plane, a line goes through (negative 2, 1) and (0, negative 5) and another line goes through (0, negative 1) and (1, 0).
Which of the following statements are true of the graphed system of equations? Check all that apply.
The system has infinitely many solutions.
A solution to the system is (–1, –2).
A solution to the system is (0, –1).
One of the equations is y = x –1.
One of the equations is 3x + y = –5.

Answer 1

To analyze the two lines provided, we need to find the equations for each line and then check the statements accordingly.

### Finding the equations of the lines:

1. **Line 1**: It passes through the points (-2, 1) and (0, -5).
- The slope (m) of a line through two points (x1, y1) and (x2, y2) is given by:
\[
m = \frac{y2 - y1}{x2 - x1} = \frac{-5 - 1}{0 - (-2)} = \frac{-6}{2} = -3
\]
- Now we can use the point-slope form of the equation of the line \( y - y_1 = m(x - x_1) \):
\[
y - 1 = -3(x + 2)
\]
- Simplifying this:
\[
y - 1 = -3x - 6 \implies y = -3x - 5
\]

So, **the equation of Line 1 is**:
\[
y = -3x - 5
\]

2. **Line 2**: It passes through the points (0, -1) and (1, 0).
- The slope (m) is:
\[
m = \frac{0 - (-1)}{1 - 0} = \frac{1}{1} = 1
\]
- Using the point-slope form:
\[
y - (-1) = 1(x - 0) \implies y + 1 = x \implies y = x - 1
\]

So, **the equation of Line 2 is**:
\[
y = x - 1
\]

### Now we can assess the statements:

1. **The system has infinitely many solutions.**
**False.** The two lines have different slopes \((-3\) for Line 1 and \(1\) for Line 2), so they will intersect at exactly one point.

2. **A solution to the system is (–1, –2).**
We can check this by substituting \(x = -1\) into both equations:
- For Line 1:
\[
y = -3(-1) - 5 = 3 - 5 = -2 \quad \text{(true)}
\]
- For Line 2:
\[
y = -1 - 1 = -2 \quad \text{(false)}
\]
Therefore, **False.**

3. **A solution to the system is (0, –1).**
For \( (0, -1) \):
- For Line 1:
\[
y = -3(0) - 5 = -5 \quad \text{(false)}
\]
- For Line 2:
\[
y = 0 - 1 = -1 \quad \text{(true)}
\]
Since it's not a solution for Line 1, **False.**

4. **One of the equations is \(y = x - 1\).**
This is indeed the equation of Line 2, so the statement is **True.**

5. **One of the equations is \(3x + y = -5\).**
The equation in a slope-intercept form for Line 1 is \(y = -3x - 5\). Converting it to standard form:
\[
3x + y + 5 = 0 \implies 3x + y = -5
\]
So this is also true, thus the statement is **True.**

### Final Results:
- The true statements are:
- One of the equations is \(y = x - 1\).
- One of the equations is \(3x + y = -5\).