Line $AB$ passes through points $A(-6,6)$ and $B(12,3)$. If the equation of the line is written in slope-intercept form, $y=mx+b$, then $m=-\frac{1}{6}$. What is the value of $b$?
Answer (2)
Substitute the given slope m = − 6 1 into the slope-intercept form: y = − 6 1 x + b .
Substitute the coordinates of point A ( − 6 , 6 ) into the equation: 6 = − 6 1 ( − 6 ) + b .
Solve for b : 6 = 1 + b ⇒ b = 5 .
The value of b is 5 .
Explanation
Understanding the Problem We are given that line A B passes through points A ( − 6 , 6 ) and B ( 12 , 3 ) . The equation of the line is given in slope-intercept form as y = m x + b , where m = − 6 1 . We need to find the value of b .
Using the Slope-Intercept Form We know the slope-intercept form of a line is y = m x + b . We are given that m = − 6 1 . So, the equation of the line is y = − 6 1 x + b . To find b , we can substitute the coordinates of either point A or point B into the equation.
Substituting Point A Let's use point A ( − 6 , 6 ) . Substituting x = − 6 and y = 6 into the equation, we get: 6 = − 6 1 ( − 6 ) + b
Simplifying the Equation Simplifying the equation, we have: 6 = 1 + b
Solving for b Subtracting 1 from both sides, we get: b = 6 − 1 = 5
Substituting Point B Alternatively, let's use point B ( 12 , 3 ) . Substituting x = 12 and y = 3 into the equation, we get: 3 = − 6 1 ( 12 ) + b
Simplifying the Equation Simplifying the equation, we have: 3 = − 2 + b
Solving for b Adding 2 to both sides, we get: b = 3 + 2 = 5
Final Answer In both cases, we find that b = 5 .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, consider a taxi service that charges a fixed fee plus a per-mile rate. If the fixed fee is b and the per-mile rate is m , the total cost y for a trip of x miles can be modeled by the equation y = m x + b . Determining the fixed fee ( b ) helps in budgeting and comparing different service options. Similarly, in physics, understanding linear relationships helps in analyzing motion with constant velocity, where the equation d = v t + d 0 represents the distance d traveled after time t , with initial distance d 0 and constant velocity v .
The value of b in the equation of the line is 5 , derived from substituting point A ( − 6 , 6 ) or point B ( 12 , 3 ) into the slope-intercept form. Both points yield the same value of b .
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