Find the equation of the line passing through the points $(1,-5)$ and $(9,11)$.
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Answer (1)
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 point ( 1 , − 5 ) to get y + 5 = 2 ( x − 1 ) .
Convert to slope-intercept form by solving for y , resulting in y = 2 x − 7 .
The equation of the line is y = 2 x − 7 .
Explanation
Problem Analysis We are given two points, ( 1 , − 5 ) and ( 9 , 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 slope-intercept form as y = m x + b , where m is the slope and b is the y-intercept.
Calculating 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.
Slope Calculation Result Plugging in the coordinates of the given points, ( 1 , − 5 ) and ( 9 , 11 ) , into the slope formula, we get: m = 9 − 1 11 − ( − 5 ) = 8 11 + 5 = 8 16 = 2 So, the slope of the line is m = 2 .
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.
Substituting Values Let's use the point ( 1 , − 5 ) . Substituting the values into the point-slope form, we get: y − ( − 5 ) = 2 ( x − 1 ) y + 5 = 2 x − 2
Converting to Slope-Intercept Form Now, we need to convert this equation to the slope-intercept form, y = m x + b . To do this, we solve for y :
y = 2 x − 2 − 5 y = 2 x − 7
Final Equation From the equation y = 2 x − 7 , we can see that the y-intercept is b = − 7 . Therefore, the equation of the line is y = 2 x − 7 .
Final Answer The equation of the line passing through the points ( 1 , − 5 ) and ( 9 , 11 ) is y = 2 x − 7 .
Examples
Understanding linear equations is crucial in many real-world applications. For example, 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 points, you can find the equation of the line and use it to predict demand at other price levels. Similarly, in physics, you can use linear equations to describe motion with constant velocity, where the equation relates time and distance. These models help in making informed decisions and predictions based on available data.
