A line has a slope of -2. It also passes through the point $(0,6)$. What is the equation of the line?
A. $y=-2 x-6$
B. $y=-2 x+6$
C. $y=6 x-2$
D. $y=6 x+2$
Answer (1)
The problem provides the slope and a point on a line and asks for the equation of the line.
Substitute the given slope m = − 2 into the slope-intercept form y = m x + b , resulting in y = − 2 x + b .
Substitute the point ( 0 , 6 ) into the equation to solve for the y-intercept b , which gives b = 6 .
Write the final equation of the line as y = − 2 x + 6 .
Explanation
Understanding the Problem We are given that a line has a slope of -2 and passes through the point (0, 6). We need to find the equation of this line.
Using Slope-Intercept Form The slope-intercept form of a linear equation is given by y = m x + b , where m is the slope and b is the y-intercept.
Substituting the Slope We are given that the slope m = − 2 . So, the equation becomes y = − 2 x + b .
Finding the y-intercept We are also given that the line passes through the point (0, 6). This means that when x = 0 , y = 6 . We can substitute these values into the equation to find b : 6 = − 2 ( 0 ) + b 6 = 0 + b b = 6
Writing the Equation of the Line Now that we have the slope m = − 2 and the y-intercept b = 6 , we can write the equation of the line as: y = − 2 x + 6
Final Answer Therefore, the equation of the line is y = − 2 x + 6 .
Examples
Understanding linear equations is crucial in many real-world applications. For example, if you are tracking the depreciation of a car, the value of the car decreases linearly over time. If the initial value of the car is $20,000 and it depreciates at a rate of 2 , 000 p erye a r , t h ee q u a t i o n re p rese n t in g t h ec a r ′ s v a l u e y a f t er x ye a rs i s y = -2000x + 20000$. This is a linear equation, and understanding how to derive and interpret such equations helps in making informed financial decisions.
