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The table shows a linear relationship between two variables, [tex]\( x \)[/tex] and [tex]\( y \)[/tex].

[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
x & 2 & 4 & 8 & 12 \\
\hline
y & 6 & 9 & 15 & 21 \\
\hline
\end{tabular}
\][/tex]

Which equation models the data in the table?

A. [tex]\( 2x - 3y = 6 \)[/tex]
B. [tex]\( 2x + 3y = 6 \)[/tex]
C. [tex]\( 2x - 3y = -6 \)[/tex]
D. [tex]\( 3x - 2y = -6 \)[/tex]

Sagot :

GinnyAnswer
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]
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