Answered

Complete the statements about finding the equation of the line that is parallel to line $n$ and passes through the point $(2, -3)$.



The slope of the graphed line is [blank].

The slope of the parallel line is [blank].



An equation that can be used to find the $y$-intercept of the parallel line is:

  • $-3 = (-2/5)(2) + b$
  • $-3 = (5/2)(2) + b$
  • $2 = (5/2)(-3) + b$
  • $2 = (2/5)(-3) + b$



The $y$-intercept of the parallel line is [blank].

The equation of the parallel line is [blank].

Verified Answer :

  • Use the slope-intercept form: \(y = mx + b\).
  • Substitute the given point \((2, -3)\) and the provided slope \(\frac{5}{2}\) into the equation.
  • Identify that \(-3 = \left(\frac{5}{2}\right)(2) + b\) is the correct expression to solve for the y-intercept.
  • Final answer: \(\boxed{-3 = \left(\frac{5}{2}\right)(2) + b}\).

Explanation

  1. Analyzing the Linear Equation and Substituting Given Values

To find the equation of a line passing through a point \((x, y) = (2, -3)\) with a given slope \(m = \frac{5}{2}\), we use the slope-intercept form of a linear equation, which is:
\[y = mx + b\]
where \(m\) is the slope and \(b\) is the y-intercept. In this problem, we are given the coordinates of the point \((2, -3)\) and the slope \(\frac{5}{2}\). We need to substitute these values into the equation to identify the correct setup for finding the y-intercept \(b\). Substituting \(y = -3\), \(x = 2\), and \(m = \frac{5}{2}\) gives:
\[-3 = \left(\frac{5}{2}\right)(2) + b\]
Comparing this result to the given options, we can see it matches the second equation provided in the list.

  1. Concluding the Analysis and Selecting the Correct Equation

The equation \(-3 = \left(\frac{5}{2}\right)(2) + b\) is the correct setup derived from the slope-intercept form using the provided point \((2, -3)\) and the specified slope of \(\frac{5}{2}\). Solving this for \(b\) would allow us to write the full equation of the line, but the task specifically asks to identify the equation that represents this step. Therefore, the second option is the correct one.
The final result is \(\boxed{-3 = \left(\frac{5}{2}\right)(2) + b}\).

Examples

In real-world applications, such as urban planning or engineering, understanding how lines interact on a coordinate plane is essential. For example, if a road is being designed with a specific grade (slope) and it must pass through a landmark location on a map, the slope-intercept form helps engineers determine the exact placement (the intercept) of that road relative to a chosen origin. This mathematical foundation allows for precise construction and spatial reasoning in complex designs.

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