A line passes through the point (-3,1) and has a slope of -2. Find an equation of this line.
Answer (1)
Use the point-slope form of a linear equation: y − y 1 = m ( x − x 1 ) .
Substitute the given point ( − 3 , 1 ) and slope − 2 into the point-slope form: y − 1 = − 2 ( x + 3 ) .
Simplify the equation to slope-intercept form: y = − 2 x − 5 .
The equation of the line is y = − 2 x − 5 .
Explanation
Understanding the Problem We are given a point that a line passes through, ( − 3 , 1 ) , and the slope of the line, − 2 . Our goal is to find the equation of this line.
Using Point-Slope Form The point-slope form of a linear equation is given by: y − y 1 = m ( x − x 1 ) where ( x 1 , y 1 ) is a point on the line and m is the slope of the line.
Substituting the Values We are given the point ( − 3 , 1 ) , so x 1 = − 3 and y 1 = 1 . We are also given the slope m = − 2 . Substituting these values into the point-slope form, we get: y − 1 = − 2 ( x − ( − 3 )) y − 1 = − 2 ( x + 3 )
Simplifying the Equation Now, we simplify the equation to slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. y − 1 = − 2 x − 6 y = − 2 x − 6 + 1 y = − 2 x − 5
Final Answer The equation of the line is y = − 2 x − 5 .
Examples
Understanding linear equations is crucial in many real-world scenarios. For example, if you are saving money, you can model your savings with a linear equation where the slope represents your rate of saving per month, and the y-intercept represents your initial savings. Similarly, in physics, the relationship between distance, speed, and time can be modeled using a linear equation, where speed is the slope and initial distance is the y-intercept. Linear equations are also used in business to model costs, revenue, and profit.
