Question

identify which equation corresponds to the following coordinates: (13, -12) and (-10, -23).
all of the answers
11x - 23y = 189
y = (11/23)x - 189/23
none of the answers

Explanation:

Step1: Find the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For points \((13, - 12)\) and \((-10, -23)\), we have \( x_1 = 13,y_1=-12,x_2=-10,y_2 = - 23 \). So \( m=\frac{-23-(-12)}{-10 - 13}=\frac{-23 + 12}{-23}=\frac{-11}{-23}=\frac{11}{23} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Using the point \((13,-12)\) and \( m=\frac{11}{23} \), we get:
\( y-(-12)=\frac{11}{23}(x - 13) \)
\( y + 12=\frac{11}{23}x-\frac{143}{23} \)
\( y=\frac{11}{23}x-\frac{143}{23}-12 \)
\( y=\frac{11}{23}x-\frac{143}{23}-\frac{276}{23} \)
\( y=\frac{11}{23}x-\frac{419}{23} \)? Wait, no, let's check the given equations. Wait, maybe we made a mistake. Wait, let's check the equation \( 11x-23y = 189 \). Let's rearrange it:
\( - 23y=-11x + 189 \)
\( y=\frac{11}{23}x-\frac{189}{23} \)
Now, let's check if the point \((13,-12)\) satisfies \( y=\frac{11}{23}x-\frac{189}{23} \)
Right - hand side: \( \frac{11\times13-189}{23}=\frac{143 - 189}{23}=\frac{-46}{23}=-2 eq - 12 \). Wait, that's a mistake. Wait, no, let's recalculate the slope. Wait, \( y_2 - y_1=-23-(-12)=-23 + 12=-11 \), \( x_2 - x_1=-10 - 13=-23 \), so slope \( m=\frac{-11}{-23}=\frac{11}{23} \). Then using point \((13,-12)\):
\( y+12=\frac{11}{23}(x - 13) \)
Multiply both sides by 23: \( 23y+276 = 11x-143 \)
\( 11x-23y=276 + 143=419 \). Wait, but the equation given is \( 11x-23y = 189 \). Wait, maybe we took the wrong point. Let's check the point \((-10,-23)\) in \( y=\frac{11}{23}x-\frac{189}{23} \)
\( \frac{11\times(-10)-189}{23}=\frac{-110 - 189}{23}=\frac{-299}{23}=-13 eq - 23 \). Wait, this is confusing. Wait, maybe there is a miscalculation. Wait, let's re - do the slope:

Slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-23-(-12)}{-10 - 13}=\frac{-11}{-23}=\frac{11}{23} \). Correct.

Now, let's use the two - point formula for the line equation: \( y - y_1=\frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \)

\( y+12=\frac{11}{23}(x - 13) \)

Multiply both sides by 23: \( 23y+276 = 11x-143 \)

\( 11x-23y=276 + 143 = 419 \)

Now, let's check the equation \( 11x-23y = 189 \). Let's see if the points satisfy it.

For \((13,-12)\): \( 11\times13-23\times(-12)=143 + 276 = 419 eq189 \)

For \((-10,-23)\): \( 11\times(-10)-23\times(-23)=-110 + 529 = 419 eq189 \)

And the equation \( y=\frac{11}{23}x-\frac{189}{23} \), when \( x = 13 \), \( y=\frac{143-189}{23}=\frac{-46}{23}=-2 eq - 12 \); when \( x=-10 \), \( y=\frac{-110 - 189}{23}=\frac{-299}{23}=-13 eq - 23 \)

Wait, but the option "All of the answers" and the other options. Wait, maybe we made a mistake in calculation. Wait, no, let's re - check the problem. Wait, the coordinates are \((13,-12)\) and \((-10,-23)\). Let's recalculate the slope: \( m=\frac{-23-(-12)}{-10 - 13}=\frac{-11}{-23}=\frac{11}{23} \). Then the equation using point - slope is \( y - (-12)=\frac{11}{23}(x - 13) \), so \( y=\frac{11}{23}x-\frac{143}{23}-12=\frac{11}{23}x-\frac{143 + 276}{23}=\frac{11}{23}x-\frac{419}{23} \). The given equation \( y=\frac{11}{23}x-\frac{189}{23} \) has a different constant term. And \( 11x-23y = 189 \) is equivalent to \( y=\frac{11}{23}x-\frac{189}{23} \). So the points do not satisfy these equations. Wait, but that can't be. Wait, maybe the question has a typo? Or maybe we made a mistake. Wait, let's check the equation \( 11x-23y = 189 \) with \( x = 13 \): \( 11\times13=143 \), \( 23y=11x - 189=143 - 189=-46 \), \( y=-2 \). Not \(-12\). With \( x=-10 \): \( 11\times(-10)=-110 \), \( 23y=11x - 189=-110 - 189=-299 \), \( y=-13 \). Not \(-23\). And the equation \( y=\frac{11}{23}x-\frac{189}{23} \) gives the same as above. So the correct answer should be "None of the answers"? But wait, maybe we made a mistake in slope calculation. Wait, \( y_2 - y_1=-23-(-12)=-11 \), \( x_2 - x_1=-10 - 13=-23 \), slope is \( \frac{11}{23} \). Correct. Then the equation is \( 11x-23y = 419 \), which is not among the options. So the answer is "None of the answers"? But wait, the option "All of the answers" is wrong, the equation \( 11x - 23y=189 \) is wrong, the equation \( y=\frac{11}{23}x-\frac{189}{23} \) is wrong. So the answer is "None of the answers"? Wait, no, wait, let's check again. Wait, maybe the coordinates are \((13, - 2)\) and \((-10,-13)\)? Then it would work. But as per the given coordinates, let's confirm:

For point \((13,-12)\) and equation \( 11x-23y = 189 \):

Left - hand side: \( 11\times13-23\times(-12)=143 + 276 = 419 eq189 \)

For point \((-10,-23)\):

Left - hand side: \( 11\times(-10)-23\times(-23)=-110 + 529 = 419 eq189 \)

For the equation \( y=\frac{11}{23}x-\frac{189}{23} \), when \( x = 13 \), \( y=\frac{143-189}{23}=\frac{-46}{23}=-2 eq - 12 \)

When \( x=-10 \), \( y=\frac{-110 - 189}{23}=\frac{-299}{23}=-13 eq - 23 \)

So none of the given equations (except "None of the answers") is correct. Wait, but the option "All of the answers" is wrong, the other two equations are wrong. So the answer is "None of the answers"? But wait, maybe I made a mistake. Wait, let's re - derive the equation.

Slope \( m=\frac{-23-(-12)}{-10 - 13}=\frac{-11}{-23}=\frac{11}{23} \)

Equation using point \((13,-12)\):

\( y+12=\frac{11}{23}(x - 13) \)

Multiply both sides by 23:

\( 23y+276 = 11x-143 \)

\( 11x-23y=276 + 143 = 419 \)

So the correct equation is \( 11x-23y = 419 \), which is not among the options. So the answer is "None of the answers".

Wait, but the option "All of the answers" is incorrect, the equation \( 11x - 23y=189 \) is incorrect, the equation \( y=\frac{11}{23}x-\frac{189}{23} \) is incorrect. So the answer is "None of the answers".

Answer:

None of the answers