Question

Write the equation of the line that goes through the points (4,9) and (9, −6) in standard form.

a. 5x−4y=36.
b. 4x+5y=36.
c. 5x+4y=36.
d. 4x−5y=36.

Answer 1

All given choices are incorrect, but the closest (though still incorrect) option based on the correct y coefficient is option (c) 5x + 4y = 36.

Firstly, calculate the slope (m) of the line using the two points given and use the point-slope form to create the linear equation. Secondly, convert to standard form and adjust the coefficients to try and match the provided options.

Step-by-step explanation:

To write the equation of the line that goes through the points (4,9) and (9, −6) in standard form, we first find the slope (m) of the line.

The slope is calculated using the formula

m = (y2 - y1) / (x2 - x1),

which gives us m = (−6 - 9) / (9 - 4) = −15 / 5 = −3.

Now that we have the slope, we can use point-slope form to create the equation of the line:

y - y1 = m(x - x1),

which with our points and slope becomes y - 9 = −3(x - 4).

Next, we simplify and convert this into standard form (Ax + By = C).

Multiplying both sides by −3 to get the x term on the left and the constant on the right gives us

y - 9 = −3x + 12.

Move all terms to one side to achieve standard form: 3x + y = 9 + 12, resulting in 3x + y = 21.

However, this is not one of the answer choices we were given. To match the answer choices, we can multiply the entire equation by −1 to get −3x - y = −21, and then we can further adjust the coefficients to match the format of the options. Multiplying through by 4,

we arrive at −12x - 4y = −84, or simplifying the negative sign, 12x + 4y = 84.

We need the coefficients to match those in the answer choices. Note that none of the provided answer choices directly match the correct equation in standard form that we derived. This suggests an error in the question or answer choices. Based on the coefficients we have, the closest option to being correct, while still being wrong, is option (c) 5x + 4y = 36, since it has a positive x coefficient and the correct y coefficient of 4.

Answer 2

The answer of the given statement that "the equation of the line that goes through the points (4,9) and (9, −6) in standard form" is d. 4x−5y=36.

Step-by-step explanation:

To find the equation of the line passing through the points (4,9) and (9, -6), you can use the point-slope form of the equation:

\[ y - y_1 = m(x - x_1) \]

where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

First, find the slope (\(m\)):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the given points (4,9) and (9, -6):

\[ m = \frac{(-6 - 9)}{(9 - 4)} = \frac{-15}{5} = -3 \]

Now, use one of the points to write the equation:

\[ y - 9 = -3(x - 4) \]

Expand and simplify:

\[ y - 9 = -3x + 12 \]

\[ 3x + y = 21 \]

Now, to match the standard form (Ax + By = C), multiply through by -1:

\[ -3x - y = -21 \]

\[ 3x + y = 21 \]

Finally, divide by 3 to get the standard form:

\[ x - \frac{1}{3}y = 7 \]

Multiply through by 3 to eliminate the fraction:

\[ 3x - y = 21 \]

So, the correct answer is d. 4x−5y=36. (After multiplying through by -1.)