Write the equation of the line that passes through the points (-1, 2) and (6, 3) in slope-intercept form.
Answer (2)
The equation of the line passing through the points (-1, 2) and (6, 3) in slope-intercept form is y = 7 1 x + 7 15 . First, the slope was calculated to be 7 1 , and then converted from point-slope form to slope-intercept form. This process involves rearranging and solving the equation until reaching the desired format.
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Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = 7 1 .
Use the point-slope form y − y 1 = m ( x − x 1 ) with the point ( − 1 , 2 ) to get y − 2 = 7 1 ( x + 1 ) .
Convert to slope-intercept form by solving for y , resulting in y = 7 1 x + 7 15 .
The equation of the line in slope-intercept form is y = 7 1 x + 7 15 .
Explanation
Understanding the Problem We are given two points, ( − 1 , 2 ) and ( 6 , 3 ) , and we want to find the equation of the line that passes through these points in slope-intercept form, which is 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 coordinates of the given points into the slope formula, we have: m = 6 − ( − 1 ) 3 − 2 = 7 1 So, the slope of the line is 7 1 .
Using Point-Slope Form Now that we have the slope, we can use the point-slope form of a linear equation, which is: y − y 1 = m ( x − x 1 ) where ( x 1 , y 1 ) is one of the given points and m is the slope. Let's use the point ( − 1 , 2 ) .
Substituting Values Substituting the point ( − 1 , 2 ) and the slope 7 1 into the point-slope form, we get: y − 2 = 7 1 ( x − ( − 1 )) y − 2 = 7 1 ( x + 1 )
Converting to Slope-Intercept Form Now, we need to transform the equation from point-slope form to slope-intercept form, which is y = m x + b . To do this, we solve for y :
y − 2 = 7 1 x + 7 1 y = 7 1 x + 7 1 + 2 y = 7 1 x + 7 1 + 7 14 y = 7 1 x + 7 15
Final Equation in Slope-Intercept Form So, the equation of the line in slope-intercept form is: y = 7 1 x + 7 15
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you are tracking the distance a car travels over time at a constant speed, you can represent this relationship with a linear equation. Similarly, in economics, linear equations can model the relationship between the price of a product and the quantity demanded. By knowing two points on the line (e.g., two time-distance data points or two price-quantity data points), you can determine the equation and make predictions about future values.
