Question

Use the table to answer the question.

x- and y-Values
x 3 7 11 15 19
y 11 8 5 2 −1

Which option is the standard form of the equation of the line going through the points in the table?

(1 point)

Answers

Answered by GPT-4o mini
To find the equation of the line that goes through the points given in the table, we can first determine the slope (m) and then use the point-slope form of a line equation to convert it to standard form.

The points given in the table are:
- (3, 11)
- (7, 8)
- (11, 5)
- (15, 2)
- (19, -1)

**Step 1: Calculate the slope (m)**
Using two points, for example, (3, 11) and (7, 8):

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 11}{7 - 3} = \frac{-3}{4}
\]

So the slope \( m = -\frac{3}{4} \).

**Step 2: Use the point-slope form**
We can use the point-slope form of the line equation, which is given by:
\[
y - y_1 = m(x - x_1)
\]
Using the point (3, 11):
\[
y - 11 = -\frac{3}{4}(x - 3)
\]

**Step 3: Distribute and rearrange to get into slope-intercept form**
\[
y - 11 = -\frac{3}{4}x + \frac{9}{4}
\]
\[
y = -\frac{3}{4}x + \frac{9}{4} + 11
\]
Convert \( 11 \) to fourths to simplify:
\[
y = -\frac{3}{4}x + \frac{9}{4} + \frac{44}{4} = -\frac{3}{4}x + \frac{53}{4}
\]

**Step 4: Convert to standard form \(Ax + By = C\)**
Multiply the entire equation by 4 to eliminate the fraction:
\[
4y = -3x + 53
\]
Rearranging gives:
\[
3x + 4y = 53
\]

Thus, the standard form of the equation of the line is:
\[
3x + 4y = 53
\]

So the answer is \( **3x + 4y = 53** \).
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