Answer :
To find the equation that models the linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given table, we will follow these steps:
1. Identify and calculate the slope [tex]\( m \)[/tex] of the linear relationship.
2. Determine the y-intercept [tex]\( b \)[/tex].
3. Formulate the equation in slope-intercept form [tex]\( y = mx + b \)[/tex].
4. Convert the slope-intercept form to the standard form [tex]\( Ax + By + C = 0 \)[/tex].
5. Compare the result with the given options.
### Step 1: Calculate the Slope [tex]\( m \)[/tex]
Given the points (2, 6) and (4, 9) from the table, we can calculate the slope [tex]\( m \)[/tex] using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values, we get:
[tex]\[ m = \frac{9 - 6}{4 - 2} = \frac{3}{2} = 1.5 \][/tex]
### Step 2: Determine the Y-intercept [tex]\( b \)[/tex]
Using the slope [tex]\( m \)[/tex] and one of the points, let's use point (2, 6), we can find the y-intercept [tex]\( b \)[/tex] using the formula:
[tex]\[ y = mx + b \][/tex]
Substitute the point (2, 6) and the slope [tex]\( m = 1.5 \)[/tex]:
[tex]\[ 6 = 1.5 \times 2 + b \][/tex]
[tex]\[ 6 = 3 + b \][/tex]
[tex]\[ b = 6 - 3 \][/tex]
[tex]\[ b = 3 \][/tex]
### Step 3: Formulate the Equation in Slope-Intercept Form
Now we have the slope [tex]\( m = 1.5 \)[/tex] and the y-intercept [tex]\( b = 3 \)[/tex]. The equation in slope-intercept form [tex]\( y = mx + b \)[/tex] is:
[tex]\[ y = 1.5x + 3 \][/tex]
### Step 4: Convert to Standard Form
To convert [tex]\( y = 1.5x + 3 \)[/tex] into the standard form [tex]\( Ax + By + C = 0 \)[/tex], we need to eliminate the fraction. Multiply the entire equation by 2:
[tex]\[ 2y = 3x + 6 \][/tex]
Rearrange terms to match the standard form:
[tex]\[ 3x - 2y = -6 \][/tex]
### Step 5: Compare with the Given Options
Finally, we compare our derived equation [tex]\( 3x - 2y = -6 \)[/tex] with the given multiple-choice options:
- A: [tex]\( 2x - 3y = 6 \)[/tex]
- B: [tex]\( 2x + 3y = 6 \)[/tex]
- C: [tex]\( 2x - 3y = -6 \)[/tex]
- D: [tex]\( 3x - 2y = -6 \)[/tex]
The correct choice that matches our equation is:
[tex]\[ \boxed{D} \][/tex]
1. Identify and calculate the slope [tex]\( m \)[/tex] of the linear relationship.
2. Determine the y-intercept [tex]\( b \)[/tex].
3. Formulate the equation in slope-intercept form [tex]\( y = mx + b \)[/tex].
4. Convert the slope-intercept form to the standard form [tex]\( Ax + By + C = 0 \)[/tex].
5. Compare the result with the given options.
### Step 1: Calculate the Slope [tex]\( m \)[/tex]
Given the points (2, 6) and (4, 9) from the table, we can calculate the slope [tex]\( m \)[/tex] using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values, we get:
[tex]\[ m = \frac{9 - 6}{4 - 2} = \frac{3}{2} = 1.5 \][/tex]
### Step 2: Determine the Y-intercept [tex]\( b \)[/tex]
Using the slope [tex]\( m \)[/tex] and one of the points, let's use point (2, 6), we can find the y-intercept [tex]\( b \)[/tex] using the formula:
[tex]\[ y = mx + b \][/tex]
Substitute the point (2, 6) and the slope [tex]\( m = 1.5 \)[/tex]:
[tex]\[ 6 = 1.5 \times 2 + b \][/tex]
[tex]\[ 6 = 3 + b \][/tex]
[tex]\[ b = 6 - 3 \][/tex]
[tex]\[ b = 3 \][/tex]
### Step 3: Formulate the Equation in Slope-Intercept Form
Now we have the slope [tex]\( m = 1.5 \)[/tex] and the y-intercept [tex]\( b = 3 \)[/tex]. The equation in slope-intercept form [tex]\( y = mx + b \)[/tex] is:
[tex]\[ y = 1.5x + 3 \][/tex]
### Step 4: Convert to Standard Form
To convert [tex]\( y = 1.5x + 3 \)[/tex] into the standard form [tex]\( Ax + By + C = 0 \)[/tex], we need to eliminate the fraction. Multiply the entire equation by 2:
[tex]\[ 2y = 3x + 6 \][/tex]
Rearrange terms to match the standard form:
[tex]\[ 3x - 2y = -6 \][/tex]
### Step 5: Compare with the Given Options
Finally, we compare our derived equation [tex]\( 3x - 2y = -6 \)[/tex] with the given multiple-choice options:
- A: [tex]\( 2x - 3y = 6 \)[/tex]
- B: [tex]\( 2x + 3y = 6 \)[/tex]
- C: [tex]\( 2x - 3y = -6 \)[/tex]
- D: [tex]\( 3x - 2y = -6 \)[/tex]
The correct choice that matches our equation is:
[tex]\[ \boxed{D} \][/tex]
