Question

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)
Responses

3x+4y=53
3 x plus 4 y equals 53

y−11=−34(x−3)
y minus 11 equals negative Start Fraction 3 over 4 End Fraction left parenthesis x minus 3 right parenthesis

y=−34x+534
y equals negative Start Fraction 3 over 4 End Fraction x plus Start Fraction 53 over 4 End Fraction

4x+3y=21

Answer 1

To determine the standard form of the equation of the line through the points given in the table, we first need to find the slope of the line and then use that to form the linear equation.

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

1. **Calculate the slope (m)** using two points from the table:
We can use points (3, 11) and (7, 8).

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

2. **Use the point-slope form** (y - y₁ = m(x - x₁)):
Using the point (3, 11):

\[
y - 11 = -\frac{3}{4}(x - 3)
\]

3. **Convert to standard form** (Ax + By = C):
Start with the point-slope form:

\[
y - 11 = -\frac{3}{4}x + \frac{9}{4}
\]

Rearranging gives:

\[
y = -\frac{3}{4}x + \frac{9}{4} + 11
\]

Convert 11 to a fraction with a common denominator of 4:

\[
11 = \frac{44}{4}
\]

Therefore:

\[
y = -\frac{3}{4}x + \frac{53}{4}
\]

To convert this to standard form (Ax + By = C), we can multiply through 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**.

Therefore, the correct response is:
**3x + 4y = 53.**