A line passes through $(1,5)$ and is parallel to $y=3 x-2$. Write its equation in slope-intercept form.
Answer (1)
The line is parallel to y = 3 x − 2 , so it has the same slope, which is 3 .
Use the point-slope form with the point ( 1 , 5 ) and slope 3 to find the equation of the line.
Substitute the point ( 1 , 5 ) into the equation y = 3 x + b to solve for b .
The equation of the line in slope-intercept form is y = 3 x + 2 .
Explanation
Understanding the Problem We are given that a line passes through the point ( 1 , 5 ) and is parallel to the line y = 3 x − 2 . Our goal is to find the equation of this line in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Finding the Slope Since the line we want to find is parallel to y = 3 x − 2 , it has the same slope. The slope of y = 3 x − 2 is 3 , so the slope of our line is also 3 . Thus, m = 3 .
Finding the y-intercept Now we know that the equation of our line is of the form y = 3 x + b . We are given that the line passes through the point ( 1 , 5 ) . We can substitute these coordinates into the equation to solve for b : 5 = 3 ( 1 ) + b 5 = 3 + b b = 5 − 3 b = 2
Writing the Equation Now we have the slope m = 3 and the y-intercept b = 2 . We can write the equation of the line in slope-intercept form: y = 3 x + 2
Examples
Understanding linear equations is crucial in many real-world scenarios. For instance, if you are tracking the cost of a taxi ride, the initial fare is the y-intercept, and the cost per mile is the slope. Similarly, in physics, the equation of motion for an object moving at a constant velocity can be represented as a linear equation, where the velocity is the slope and the initial position is the y-intercept. Linear equations also help in simple financial planning, such as calculating savings over time with a constant deposit rate.
