What is the equation of the line that passes through the points (-3,-2) and (1, 6)?
Answer (2)
The equation of the line that passes through the points (-3, -2) and (1, 6) is given by the equation y = 2x + 4. This was found by first calculating the slope, then using the point-slope formula, and finally converting to the slope-intercept form. Thus, the final equation of the line is y = 2x + 4.
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Calculate the slope m using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = 1 − ( − 3 ) 6 − ( − 2 ) = 2 .
Use the point-slope form of the line y − y 1 = m ( x − x 1 ) with the point ( − 3 , − 2 ) to get y + 2 = 2 ( x + 3 ) .
Convert to slope-intercept form by solving for y , resulting in y = 2 x + 4 .
The equation of the line is y = 2 x + 4 .
Explanation
Understanding the Problem We are given two points, (-3, -2) and (1, 6), 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, denoted by m , is calculated as the change in y divided by the change in x . Using the formula: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) = ( − 3 , − 2 ) and ( x 2 , y 2 ) = ( 1 , 6 ) .
Substituting the given values, we have: m = 1 − ( − 3 ) 6 − ( − 2 ) = 1 + 3 6 + 2 = 4 8 = 2 So, the slope of the line is 2.
Finding the Equation of the Line 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 ) We can use either of the given points. Let's use the point (-3, -2). Substituting the values, we get: y − ( − 2 ) = 2 ( x − ( − 3 )) y + 2 = 2 ( x + 3 ) y + 2 = 2 x + 6 Now, we solve for y to get the slope-intercept form: y = 2 x + 6 − 2 y = 2 x + 4 Thus, the equation of the line in slope-intercept form is y = 2 x + 4 .
Final Answer The equation of the line that passes through the points (-3, -2) and (1, 6) is y = 2 x + 4 .
Examples
Imagine you are tracking the growth of a plant. At week 3, it was 2 inches tall, and by week 7, it had grown to 6 inches. Finding the equation of the line that represents this growth helps you predict its height at any given week, assuming the growth is linear. This is a practical application of linear equations in understanding and predicting real-world phenomena.
