Write the equation of a line that passes through $(\frac{1}{3}, 7)$ and is parallel to $y=3 x-4$.
Answer (1)
Determine the slope of the given line: The slope of y = 3 x − 4 is 3 .
Use the point-slope form: y − y 1 = m ( x − x 1 ) with the point ( 3 1 , 7 ) and slope 3 .
Substitute the values: y − 7 = 3 ( x − 3 1 ) .
Simplify to slope-intercept form: y = 3 x + 6 . The final answer is y = 3 x + 6 .
Explanation
Understanding the Problem We are given a point ( 3 1 , 7 ) and a line y = 3 x − 4 . We need to find the equation of a line that passes through the given point and is parallel to the given line.
Finding the Slope Parallel lines have the same slope. The given line is in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. The slope of the given line y = 3 x − 4 is 3 . Therefore, the slope of the line we want to find is also 3 .
Using Point-Slope Form We will use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line. We have the point ( 3 1 , 7 ) and the slope m = 3 . Substituting these values into the point-slope form, we get:
y − 7 = 3 ( x − 3 1 )
Simplifying the Equation Now, we simplify the equation to get it into slope-intercept form, y = m x + b :
y − 7 = 3 x − 3 ( 3 1 )
y − 7 = 3 x − 1
y = 3 x − 1 + 7
y = 3 x + 6
Final Answer The equation of the line that passes through ( 3 1 , 7 ) and is parallel to y = 3 x − 4 is y = 3 x + 6 .
Examples
Understanding linear equations is crucial in many real-world scenarios. For instance, imagine you are tracking the cost of a taxi ride. The initial fare is $6, and for every mile, it costs an additional $3. The equation y = 3 x + 6 models this situation, where y is the total cost, and x is the number of miles traveled. This equation helps you predict the total cost of your ride based on the distance. Similarly, in business, linear equations can model revenue, cost, and profit, aiding in financial planning and decision-making.
