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Sagot :
Let's analyze the given data points and determine a detailed, step-by-step solution for finding the missing values as well as the slope of the line that fits these points.
### Step 1: Identify Given Points
We are given pairs of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
- [tex]\( (10, 12) \)[/tex]
- [tex]\( (20, 30) \)[/tex]
Our task is to find the slope of the line passing through these two points, the y-intercept, and the missing y-value for [tex]\( x = 8 \)[/tex].
### Step 2: Calculate the Slope (m)
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the values from the points [tex]\( (10, 12) \)[/tex] and [tex]\( (20, 30) \)[/tex]:
[tex]\[ m = \frac{30 - 12}{20 - 10} \][/tex]
[tex]\[ m = \frac{18}{10} \][/tex]
[tex]\[ m = 1.8 \][/tex]
### Step 3: Find the Y-Intercept (b)
Using the slope [tex]\( m \)[/tex] and one of the points, we can find the y-intercept [tex]\( b \)[/tex]. The equation of the line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
Using the point [tex]\( (10, 12) \)[/tex], substitute [tex]\( x = 10 \)[/tex], [tex]\( y = 12 \)[/tex], and [tex]\( m = 1.8 \)[/tex]:
[tex]\[ 12 = 1.8 \cdot 10 + b \][/tex]
[tex]\[ 12 = 18 + b \][/tex]
[tex]\[ b = 12 - 18 \][/tex]
[tex]\[ b = -6 \][/tex]
So, the y-intercept is [tex]\( b = -6 \)[/tex].
### Step 4: Determine the Missing Y-Value
We need to find the missing y-value for [tex]\( x = 8 \)[/tex]. Using the equation of the line [tex]\( y = 1.8x - 6 \)[/tex] (with [tex]\( m = 1.8 \)[/tex] and [tex]\( b = -6 \)[/tex]):
[tex]\[ y = 1.8 \cdot 8 - 6 \][/tex]
[tex]\[ y = 14.4 - 6 \][/tex]
[tex]\[ y = 8.4 \][/tex]
### Summary of Results
1. Slope (m): 1.8
2. Y-Intercept (b): -6
3. Missing Y-Value for [tex]\( x = 8 \)[/tex]: 8.4
Therefore, the missing y-value for [tex]\( x = 8 \)[/tex] is 8.4, the slope of the line is 1.8, and the y-intercept is -6.
### Step 1: Identify Given Points
We are given pairs of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
- [tex]\( (10, 12) \)[/tex]
- [tex]\( (20, 30) \)[/tex]
Our task is to find the slope of the line passing through these two points, the y-intercept, and the missing y-value for [tex]\( x = 8 \)[/tex].
### Step 2: Calculate the Slope (m)
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the values from the points [tex]\( (10, 12) \)[/tex] and [tex]\( (20, 30) \)[/tex]:
[tex]\[ m = \frac{30 - 12}{20 - 10} \][/tex]
[tex]\[ m = \frac{18}{10} \][/tex]
[tex]\[ m = 1.8 \][/tex]
### Step 3: Find the Y-Intercept (b)
Using the slope [tex]\( m \)[/tex] and one of the points, we can find the y-intercept [tex]\( b \)[/tex]. The equation of the line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
Using the point [tex]\( (10, 12) \)[/tex], substitute [tex]\( x = 10 \)[/tex], [tex]\( y = 12 \)[/tex], and [tex]\( m = 1.8 \)[/tex]:
[tex]\[ 12 = 1.8 \cdot 10 + b \][/tex]
[tex]\[ 12 = 18 + b \][/tex]
[tex]\[ b = 12 - 18 \][/tex]
[tex]\[ b = -6 \][/tex]
So, the y-intercept is [tex]\( b = -6 \)[/tex].
### Step 4: Determine the Missing Y-Value
We need to find the missing y-value for [tex]\( x = 8 \)[/tex]. Using the equation of the line [tex]\( y = 1.8x - 6 \)[/tex] (with [tex]\( m = 1.8 \)[/tex] and [tex]\( b = -6 \)[/tex]):
[tex]\[ y = 1.8 \cdot 8 - 6 \][/tex]
[tex]\[ y = 14.4 - 6 \][/tex]
[tex]\[ y = 8.4 \][/tex]
### Summary of Results
1. Slope (m): 1.8
2. Y-Intercept (b): -6
3. Missing Y-Value for [tex]\( x = 8 \)[/tex]: 8.4
Therefore, the missing y-value for [tex]\( x = 8 \)[/tex] is 8.4, the slope of the line is 1.8, and the y-intercept is -6.
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