To analyze and understand the given equation of the line [tex]\( y = -6x + 3 \)[/tex], we need to break down its components step-by-step.
### Step 1: Identify the Slope and Y-Intercept
The equation [tex]\( y = -6x + 3 \)[/tex] is in the slope-intercept form [tex]\( y = mx + b \)[/tex]. In this form:
- [tex]\( m \)[/tex] represents the slope of the line.
- [tex]\( b \)[/tex] represents the y-intercept of the line, which is the point where the line crosses the y-axis.
From the given equation, we can identify:
- The slope ([tex]\( m \)[/tex]) is [tex]\(-6\)[/tex].
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\(3\)[/tex].
### Step 2: Interpret the Slope
The slope of a line indicates how steep the line is and the direction it goes. A slope of [tex]\(-6\)[/tex] means that for every unit increase in [tex]\( x \)[/tex]:
- The value of [tex]\( y \)[/tex] decreases by 6 units.
This steep negative slope means the line is falling quickly as we move from left to right.
### Step 3: Interpret the Y-Intercept
The y-intercept is the point at which the line crosses the y-axis.
- When [tex]\( x = 0 \)[/tex], the value of [tex]\( y \)[/tex] is [tex]\(3\)[/tex].
This means the point [tex]\((0, 3)\)[/tex] is on the line.
### Step 4: Sketch a Visual (Optional but Helpful)
If you were to draw the graph of this equation:
1. Start by plotting the y-intercept [tex]\((0, 3)\)[/tex] on the coordinate plane.
2. From this point, use the slope to find another point. Since the slope is [tex]\(-6\)[/tex], from [tex]\((0, 3)\)[/tex]:
- Move 1 unit to the right (positive x-direction), and 6 units down (negative y-direction), landing at the point [tex]\((1, -3)\)[/tex].
### Step 5: Confirm with Another Point (Verification)
To ensure the accuracy, let's plug in another value for [tex]\( x \)[/tex] and solve for [tex]\( y \)[/tex]:
For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -6(2) + 3 = -12 + 3 = -9 \][/tex]
Therefore, the point [tex]\((2, -9)\)[/tex] should also lie on the line.
### Conclusion
The detailed breakdown of the linear equation [tex]\( y = -6x + 3 \)[/tex] shows that:
- Slope: [tex]\(-6\)[/tex]
- Y-intercept: [tex]\( 3\)[/tex]
These components provide us with a complete understanding of the line's behavior and its graphical representation.
