Sure! Let's analyze the equation [tex]\( y = -\frac{2}{3}x + 1 \)[/tex] step-by-step to understand its components.
### Step 1: Identify the Slope
The given equation is in the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
In the equation [tex]\( y = -\frac{2}{3}x + 1 \)[/tex]:
- The slope ([tex]\( m \)[/tex]) is [tex]\( -\frac{2}{3} \)[/tex].
### Step 2: Identify the Y-Intercept
The y-intercept ([tex]\( b \)[/tex]) is the constant term present in the equation, which is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
From [tex]\( y = -\frac{2}{3}x + 1 \)[/tex]:
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\( 1 \)[/tex]. This means the line crosses the y-axis at [tex]\( y = 1 \)[/tex].
### Step 3: Plot the Y-Intercept
On the coordinate plane, you will start by plotting the y-intercept:
- Locate [tex]\( (0, 1) \)[/tex] on the graph, and mark this point.
### Step 4: Plot Another Point Using the Slope
To plot another point using the slope, use the slope value of [tex]\( -\frac{2}{3} \)[/tex]:
- The slope [tex]\( -\frac{2}{3} \)[/tex] means that for every 3 units you move to the right on the x-axis, you move down 2 units on the y-axis (since the slope is negative).
Starting from the y-intercept [tex]\( (0, 1) \)[/tex]:
- Move 3 units to the right along the x-axis, reaching [tex]\( x = 3 \)[/tex].
- From [tex]\( x = 3 \)[/tex], move 2 units down along the y-axis, reaching [tex]\( y = -1 \)[/tex].
The new point is [tex]\( (3, -1) \)[/tex].
### Step 5: Draw the Line
Now, draw a straight line passing through the two points [tex]\( (0, 1) \)[/tex] and [tex]\( (3, -1) \)[/tex].
### Summary
- The slope of the line is [tex]\( -\frac{2}{3} \)[/tex].
- The y-intercept is [tex]\( 1 \)[/tex].
- Points plotted are [tex]\( (0, 1) \)[/tex] and [tex]\( (3, -1) \)[/tex].
By following these steps, you can correctly graph the linear equation [tex]\( y = -\frac{2}{3}x + 1 \)[/tex].
