What is the equation of the line that has a slope of -7 and passes through the point (4, 8)?
Answer (2)
The equation of the line with a slope of -7 that passes through the point (4, 8) is given by the equation y = − 7 x + 36 . This is derived using the point-slope form of a linear equation. The final equation represents the relationship between x and y for that line.
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Use the point-slope form of a line: y − y 1 = m ( x − x 1 ) .
Substitute the given slope m = − 7 and the point ( 4 , 8 ) into the point-slope form: y − 8 = − 7 ( x − 4 ) .
Simplify the equation: y − 8 = − 7 x + 28 .
Convert to slope-intercept form: y = − 7 x + 36 . The equation of the line is y = − 7 x + 36 .
Explanation
Understanding the Problem We are given the slope of a line, which is m = − 7 , and a point that the line passes through, which is ( 4 , 8 ) . Our goal is to find the equation of this line.
Using Point-Slope Form We can use the point-slope form of a line, which is given by the equation: y − y 1 = m ( x − x 1 ) where m is the slope and ( x 1 , y 1 ) is a point on the line.
Substituting Values Substitute the given values into the point-slope form: y − 8 = − 7 ( x − 4 )
Simplifying the Equation Now, we can simplify the equation to slope-intercept form ( y = m x + b ) by distributing the -7 and isolating y : y − 8 = − 7 x + 28 Add 8 to both sides: y = − 7 x + 28 + 8 y = − 7 x + 36
Final Answer The equation of the line is y = − 7 x + 36 .
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 $36,000 and it depreciates at a rate of 7 , 000 p erye a r , t h ee q u a t i o n re p rese n t in g t h ec a r ′ s v a l u e y a f t er x ye a rs i s y = -7000x + 36000$. This equation helps you predict the car's value at any point in time.
