Determine the Equation of a Line Parallel to y=3x+4 Through (1/2,1)

Question

Given the line parallel to the line y=3x+4 y=3x+4

and passes through the point (12,1) (\frac{1}{2},1) .

Which of the algebraic representations is the corresponding one for the given line?

Video Solution

Solution Steps

00:00 Find the algebraic representation of the function
00:03 This is the slope of the line
00:08 Parallel lines have identical slopes
00:11 We'll use the line equation
00:15 We'll substitute the point according to the given data
00:19 We'll substitute the line's slope according to the given data
00:22 We'll continue solving to find the intersection point
00:31 We'll isolate the intersection point (B)
00:38 This is the intersection point with the Y-axis
00:43 Now we'll substitute the intersection point and slope in the line equation
00:55 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we begin by noting that since the line is parallel to y=3x+4 y = 3x + 4 , it must have the same slope, m=3 m = 3 .

We use the point-slope form of the equation of a line, which is:

yy1=m(xx1) y - y_1 = m(x - x_1)

Here, the slope m=3 m = 3 and the line passes through the point (12,1) \left(\frac{1}{2}, 1\right) . Therefore, we substitute these values into the point-slope formula:

y1=3(x12) y - 1 = 3\left(x - \frac{1}{2}\right)

Next, we simplify this equation:

  • Distribute the slope 3 3 on the right side:
  • y1=3x32 y - 1 = 3x - \frac{3}{2}
  • Add 1 to both sides to solve for y y :
  • y=3x32+1 y = 3x - \frac{3}{2} + 1
  • Simplify 32+1-\frac{3}{2} + 1:
  • y=3x12 y = 3x - \frac{1}{2}

Thus, the equation of the line parallel to y=3x+4 y = 3x + 4 and passing through the point (12,1) \left(\frac{1}{2}, 1\right) is:

y=3x12 y = 3x - \frac{1}{2}

The corresponding choice is:

y=3x12 y=3x-\frac{1}{2}

Answer

y=3x12 y=3x-\frac{1}{2}