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Algebra
Pregunta

7. Determine what rule the function machine is usi...

7. Determine what rule the function machine is using. 

Input           Output 

\(49\)               \(44\)

\(38\)               \(33\)

\(24\)               \(19\)

\(16\)               \(11\)

Answer

Your input: find the equation of a line given two points    \( P = ( 49,44 ) \) and \( Q = ( 38,33 ) \) . The slope of a line passing through the two points \( P = ( x _ { 1 } , y _ { 1 } ) \) and \( Q = ( x _ { 2 } , y _ { 2 } ) \) is given by \( m = \frac { y _ { 2 } - y _ { 1 } } { x _ { 2 } - x _ { 1 } }\) 

We have that \( x _ { 1 } = 49 , y _ { 1 } = 44 , x _ { 2 } = 38 , y _ { 2 } = 33 \) 

Plug the given values into the formula for slope: \( m = \frac { ( 33 ) - ( 44 ) } { ( 38 ) - ( 49 ) } = \frac { - 11 } { - 11 } = 1\) 

Now, the \( y \) -intercept is \( b = y _ { 1 } - m \cdot x _ { 1 } \) (or \( b = y _ { 2 } - m \cdot x _ { 2 } \) , the result is the same). 

\( b = 44 - ( 1 ) \cdot ( 49 ) = - 5 \) . 

Finally, the equation of the line can be written in the form \( y = m x + b \) . 

\(y = x - 5 \) 

Answer: 

The slope of the line is \( m = 1\) 

The equation of the line in the slope-intercept form is \( y = x - 5 \) . 

The equation of the line in the point-slope form is \( y - 44 = 1 ( x - 49 ) \) . 

The equation of the line in the point-slope form is \( y - 33 = 1 ( x - 38 ) \) . 

The general equation of the line is \( x - y - 5 = 0 \) . 

Solución
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