A line has a slope of -2. It also passes through the point $(0,6)$. What is the equation of the line?
A. $y=6 x-2$
B. $y=6 x+2$
C. $y=-2 x+6$
D. $y=-2 x-6$
Answer (1)
The problem provides the slope and a point on a line and asks for the equation of the line.
Use the slope-intercept form of a line, y = m x + b , where m is the slope and b is the y-intercept.
Substitute the given slope m = − 2 and the point ( 0 , 6 ) into the equation to solve for b .
The equation of the line is 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 .
Substituting the Point We are also given that the line passes through the point (0, 6). We can substitute these coordinates into the equation to find the y-intercept b . So, 6 = − 2 ( 0 ) + b .
Finding the y-intercept Simplifying the equation, we get 6 = 0 + b , which means b = 6 .
Writing the Equation of the Line Now we can write the complete equation of the line by substituting the values of m and b into the slope-intercept form: 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 instance, 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 $20,000 and it depreciates at a rate of 2 , 000 p erye a r , t h ee q u a t i o n re p rese n t in g t h ec a r ′ s v a l u e y a f t er x ye a rs w o u l d b e y = -2000x + 20000$. This equation allows you to predict the car's value at any point in time. Similarly, linear equations are used in calculating simple interest, where the amount of interest earned is a linear function of the principal amount and the interest rate.
