Write the point-slope form of the line satisfying the given conditions. Then use the point-slope form to write the slope-intercept form of the equation. Slope = 4, passing through (-7,3)
Answer (2)
Write the point-slope form using the formula y − y 1 = m ( x − x 1 ) with m = 4 and ( x 1 , y 1 ) = ( − 7 , 3 ) , which gives y − 3 = 4 ( x + 7 ) .
Convert the point-slope form to slope-intercept form by distributing the 4 and isolating y: y − 3 = 4 x + 28 .
Add 3 to both sides to get y = 4 x + 31 .
The point-slope form is y − 3 = 4 ( x + 7 ) and the slope-intercept form is y = 4 x + 31 .
Explanation
Understanding the Problem We are given the slope of a line, m = 4 , and a point that the line passes through, ( − 7 , 3 ) . Our goal is to find the equation of this line in both point-slope form and slope-intercept form.
Point-Slope Form The point-slope form of a line 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 m = 4 and ( x 1 , y 1 ) = ( − 7 , 3 ) into the point-slope form: y − 3 = 4 ( x − ( − 7 )) Simplify the equation: y − 3 = 4 ( x + 7 ) This is the point-slope form of the equation of the line.
Slope-Intercept Form Now, we want to convert the point-slope form to slope-intercept form, which is given by the equation: y = m x + b where m is the slope and b is the y-intercept.
Converting to Slope-Intercept Form Starting from the point-slope form: y − 3 = 4 ( x + 7 ) Distribute the 4 on the right side: y − 3 = 4 x + 28 Add 3 to both sides to isolate y: y = 4 x + 28 + 3 Simplify: y = 4 x + 31 This is the slope-intercept form of the equation of the line.
Final Answer The point-slope form of the equation of the line is y − 3 = 4 ( x + 7 ) , and the slope-intercept form of the equation of the line is y = 4 x + 31 .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you are tracking the cost of a taxi ride, the point-slope form can help you determine the total fare based on a fixed initial charge and a per-mile rate. Similarly, in physics, you can use linear equations to describe the motion of an object with constant velocity. These equations also play a vital role in economics, where they can model supply and demand curves, helping businesses make informed decisions about pricing and production.
The point-slope form of the line is y − 3 = 4 ( x + 7 ) and the slope-intercept form is y = 4 x + 31 .
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