Find the equation of the line passing through the points $(5,21)$ and (-5,-29).
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Answer (2)
Calculate the slope using the formula: m = x 2 − x 1 y 2 − y 1 = − 5 − 5 − 29 − 21 = 5 .
Use the point-slope form with point ( 5 , 21 ) : y − 21 = 5 ( x − 5 ) .
Convert to slope-intercept form: y = 5 x − 25 + 21 .
The equation of the line is: y = 5 x − 4 .
Explanation
Understanding the Problem We are given two points, ( 5 , 21 ) and ( − 5 , − 29 ) , and we want to find the equation of the line that passes through them. The equation of a line can be written in the slope-intercept form as y = m x + b , where m is the slope and b is the y-intercept.
Finding the Slope First, we need to find the slope m of the line. The slope is defined as the change in y divided by the change in x , which can be calculated using 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 given points. In our case, ( x 1 , y 1 ) = ( 5 , 21 ) and ( x 2 , y 2 ) = ( − 5 , − 29 ) .
Calculating the Slope Substituting the coordinates of the points into the slope formula, we get: m = − 5 − 5 − 29 − 21 = − 10 − 50 = 5 So, the slope of the line is m = 5 .
Using Point-Slope Form Now that we have the slope, we can use the point-slope form of a line to find the equation of the line. The point-slope form is given by: y − y 1 = m ( x − x 1 ) where ( x 1 , y 1 ) is one of the given points and m is the slope. We can use the point ( 5 , 21 ) .
Finding the Equation Substituting the slope m = 5 and the point ( 5 , 21 ) into the point-slope form, we get: y − 21 = 5 ( x − 5 ) Now, we need to rewrite this equation in the slope-intercept form, y = m x + b .
Simplifying the Equation Expanding the equation, we have: y − 21 = 5 x − 25 Adding 21 to both sides, we get: y = 5 x − 25 + 21 y = 5 x − 4 So, the equation of the line is y = 5 x − 4 .
Final Answer The equation of the line is y = 5 x − 4 . Therefore, the slope is 5 and the y-intercept is -4.
Examples
Understanding linear equations is crucial in many real-world applications. For instance, in economics, you might use a linear equation to model the relationship between the price of a product and the quantity demanded. If you know two price-quantity data points, you can determine the equation of the line and predict demand at other price points. Similarly, in physics, you can describe the motion of an object moving at a constant velocity using a linear equation, where the slope represents the velocity and the y-intercept represents the initial position. Linear equations are also used in computer graphics to draw lines and shapes on the screen.
The equation of the line passing through the points (5, 21) and (-5, -29) is given by y = 5x - 4. This is derived by calculating the slope and using the point-slope form of a line. The final equation indicates a slope of 5 and a y-intercept of -4.
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