Question
A line is defined by the equation y = negative x + 3. Which shows the graph of this line?
On a coordinate plane, a line goes through points (0, 3) and (3, 0).
On a coordinate plane, a line goes through points (0, negative 3) and (0, 3).
On a coordinate plane, a line goes through points (0, 0) and (2, 6).
On a coordinate plane, a line goes through points (negative 2, 6) and (0, 0).
On a coordinate plane, a line goes through points (0, 3) and (3, 0).
On a coordinate plane, a line goes through points (0, negative 3) and (0, 3).
On a coordinate plane, a line goes through points (0, 0) and (2, 6).
On a coordinate plane, a line goes through points (negative 2, 6) and (0, 0).
Answers
GPT-4o mini
The equation of the line is given as \( y = -x + 3 \). This is in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
From the equation \( y = -x + 3 \):
- The slope (\( m \)) is \(-1\).
- The y-intercept (\( b \)) is \(3\), meaning the line crosses the y-axis at the point \((0, 3)\).
To find another point on the line, we can choose \( x = 3 \):
\[
y = -3 + 3 = 0
\]
So, when \( x = 3 \), \( y = 0\), which gives us the point \((3, 0)\).
Thus, the line passes through the points \((0, 3)\) and \((3, 0)\).
Now, we will examine the given options:
1. **On a coordinate plane, a line goes through points (0, 3) and (3, 0).** ✔️ This matches the points we found on the line.
2. **On a coordinate plane, a line goes through points (0, negative 3) and (0, 3).** ❌ The y-intercept is \(3\), not \(-3\).
3. **On a coordinate plane, a line goes through points (0, 0) and (2, 6).** ❌ This line does not have the correct slope or intercept.
4. **On a coordinate plane, a line goes through points (negative 2, 6) and (0, 0).** ❌ This line also does not match the equation given.
Therefore, the correct graph of the line defined by \( y = -x + 3 \) is the first option: **On a coordinate plane, a line goes through points (0, 3) and (3, 0).**
From the equation \( y = -x + 3 \):
- The slope (\( m \)) is \(-1\).
- The y-intercept (\( b \)) is \(3\), meaning the line crosses the y-axis at the point \((0, 3)\).
To find another point on the line, we can choose \( x = 3 \):
\[
y = -3 + 3 = 0
\]
So, when \( x = 3 \), \( y = 0\), which gives us the point \((3, 0)\).
Thus, the line passes through the points \((0, 3)\) and \((3, 0)\).
Now, we will examine the given options:
1. **On a coordinate plane, a line goes through points (0, 3) and (3, 0).** ✔️ This matches the points we found on the line.
2. **On a coordinate plane, a line goes through points (0, negative 3) and (0, 3).** ❌ The y-intercept is \(3\), not \(-3\).
3. **On a coordinate plane, a line goes through points (0, 0) and (2, 6).** ❌ This line does not have the correct slope or intercept.
4. **On a coordinate plane, a line goes through points (negative 2, 6) and (0, 0).** ❌ This line also does not match the equation given.
Therefore, the correct graph of the line defined by \( y = -x + 3 \) is the first option: **On a coordinate plane, a line goes through points (0, 3) and (3, 0).**