Find the equation of the line passing through the points $(7,20)$ and $(-2,11)$.
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Answer (1)
Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 : m = − 2 − 7 11 − 20 = 1 .
Use the point-slope form with the point ( 7 , 20 ) : y − 20 = 1 ( x − 7 ) .
Convert to slope-intercept form: y = x − 7 + 20 .
The equation of the line is y = x + 13 .
Explanation
Understanding the Problem We are given two points, ( 7 , 20 ) and ( − 2 , 11 ) , and we want to find the equation of the line that passes through them. The equation of a line can be written in the form 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 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.
Calculating the Slope Substituting the given points ( 7 , 20 ) and ( − 2 , 11 ) into the slope formula, we get: m = − 2 − 7 11 − 20 = − 9 − 9 = 1 So, the slope of the line is m = 1 .
Using Point-Slope Form Now that we have the slope, we can use the point-slope form of a line to find the equation. The point-slope form is: y − y 1 = m ( x − x 1 ) where ( x 1 , y 1 ) is one of the given points. Let's use the point ( 7 , 20 ) .
Converting to Slope-Intercept Form Substituting m = 1 and the point ( 7 , 20 ) into the point-slope form, we get: y − 20 = 1 ( x − 7 ) Now, we can rewrite this equation in slope-intercept form ( y = m x + b ) by solving for y :
Finding the Equation of the Line y − 20 = x − 7 y = x − 7 + 20 y = x + 13 So, the equation of the line is y = x + 13 .
Final Answer The equation of the line passing through the points ( 7 , 20 ) and ( − 2 , 11 ) is y = x + 13 . Therefore, the slope is 1 and the y-intercept is 13.
Examples
Understanding linear equations is crucial in many real-world applications. For instance, consider a taxi service that charges a fixed fee plus a per-mile rate. If the taxi charges $20 for a 7-mile ride and $11 for a 2-mile ride, we can use the points (7, 20) and (2, 11) to determine the per-mile rate (slope) and the fixed fee (y-intercept). This allows us to predict the cost of any ride length, demonstrating the practical use of linear equations in everyday scenarios.
