Find the equation of the line specified.
The line passes through the points $(7,-7)$ and $(6,-5)$.
a. $y=-2 x+7$
b. $y=2 x-21$
c. $y=-2 x-7$
d. $y=2 x-7$
Answer (2)
Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = − 2 .
Use the point-slope form of the line y − y 1 = m ( x − x 1 ) with the point ( 7 , − 7 ) to get y + 7 = − 2 ( x − 7 ) .
Convert to slope-intercept form by simplifying the equation to y = − 2 x + 14 − 7 .
The final equation of the line is y = − 2 x + 7 , so the answer is y = − 2 x + 7 .
Explanation
Understanding the Problem We are given two points, ( 7 , − 7 ) and ( 6 , − 5 ) , and we want to find the equation of the line that passes through them.
Calculating the Slope First, we need to find the slope of the line. The slope, m , is given by the formula: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two points. In our case, ( x 1 , y 1 ) = ( 7 , − 7 ) and ( x 2 , y 2 ) = ( 6 , − 5 ) . Plugging these values into the formula, we get: m = 6 − 7 − 5 − ( − 7 ) = − 1 − 5 + 7 = − 1 2 = − 2 So, the slope of the line is − 2 .
Using the Point-Slope Form Now that we have the slope, we can use the point-slope form of a line, which is given by: y − y 1 = m ( x − x 1 ) where ( x 1 , y 1 ) is a point on the line and m is the slope. We can use either of the given points. Let's use ( 7 , − 7 ) . Plugging in the values, we get: y − ( − 7 ) = − 2 ( x − 7 ) y + 7 = − 2 x + 14
Converting to Slope-Intercept Form Now, we need to convert this equation to the slope-intercept form, which is y = m x + b . To do this, we simply subtract 7 from both sides of the equation: y = − 2 x + 14 − 7 y = − 2 x + 7 So, the equation of the line is y = − 2 x + 7 .
Final Answer Comparing our equation with the given options, we see that it matches option a. Therefore, the correct answer is y = − 2 x + 7 .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you are tracking the depreciation of a car's value over time, you might find that it decreases linearly. If the car was initially worth $20,000 and depreciates by 2 , 000 e a c h ye a r , t h ec a r ′ s v a l u e y a f t er x ye a rsc anb e m o d e l e d b y t h ee q u a t i o n y = -2000x + 20000$. This allows you to predict the car's value at any point in time. Similarly, linear equations are used in physics to describe motion with constant velocity, in economics to model supply and demand curves, and in many other fields to represent relationships between two variables.
The equation of the line passing through the points (7,-7) and (6,-5) is y = − 2 x + 7 . Among the provided options, the correct choice is (a) y = − 2 x + 7 . This was determined by calculating the slope and using point-slope form to derive the equation.
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