1. Find the equation of a line which is parallel to [tex]y=3x-7[/tex] and passes through the point [tex]P(4,2)[/tex].
2. Line [tex]L_1[/tex] which passes through points [tex]A(1,0)[/tex] and [tex]B(2,2)[/tex] is parallel to line [tex]L_2[/tex] which passes through points [tex]C(1,3)[/tex] and [tex]D(k, 5)[/tex]. Determine the value of [tex]k[/tex].
3. [tex]P[/tex], [tex]Q[/tex], [tex]R[/tex], and [tex]S[/tex] are at points [tex](-2,4)[/tex], [tex](3,-1)[/tex], [tex](n, n)[/tex] and [tex](5,1)[/tex] respectively. If [tex]PQ[/tex] and [tex]RS[/tex] are parallel, find the value of [tex]n[/tex].

Determine the equation of line d using the point-slope form and the fact that parallel lines have the same slope: y = 3 x − 10 .
Calculate the slope of lines L 1 ​ and L 2 ​ , equate them, and solve for k : k = 2 .
Calculate the slope of lines PQ and RS , equate them, and solve for n : n = 3 .
The equation of line d is y = 3 x − 10 , the value of k is 2 , and the value of n is 3 : y = 3 x − 10 ; k = 2 ; n = 3 ​ .

Explanation

Problem Analysis The problem provides three separate questions related to parallel lines. We need to find the equation of a line parallel to a given line, find the value of k such that two lines are parallel, and find the value of n such that two lines are parallel.

Finding the equation of line d The equation of the given line is y = 3 x − 7 . Since parallel lines have the same slope, the slope of line d is 3 . Using the point-slope form of a line, 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 y − 2 = 3 ( x − 4 ) . Simplifying, we get y − 2 = 3 x − 12 , so y = 3 x − 10 .

Finding the value of k Line L 1 ​ passes through points A ( 1 , 0 ) and B ( 2 , 2 ) . The slope of L 1 ​ is m 1 ​ = 2 − 1 2 − 0 ​ = 1 2 ​ = 2 . Line L 2 ​ passes through points C ( 1 , 3 ) and D ( k , 5 ) . The slope of L 2 ​ is m 2 ​ = k − 1 5 − 3 ​ = k − 1 2 ​ . Since L 1 ​ and L 2 ​ are parallel, their slopes are equal, so m 1 ​ = m 2 ​ . Thus, 2 = k − 1 2 ​ . Multiplying both sides by k − 1 , we get 2 ( k − 1 ) = 2 . Dividing both sides by 2, we get k − 1 = 1 , so k = 2 .

Finding the value of n Line PQ passes through points P ( − 2 , 4 ) and Q ( 3 , − 1 ) . The slope of PQ is m PQ ​ = 3 − ( − 2 ) − 1 − 4 ​ = 5 − 5 ​ = − 1 . Line RS passes through points R ( n , n ) and S ( 5 , 1 ) . The slope of RS is m RS ​ = 5 − n 1 − n ​ . Since PQ and RS are parallel, their slopes are equal, so m PQ ​ = m RS ​ . Thus, − 1 = 5 − n 1 − n ​ . Multiplying both sides by 5 − n , we get − ( 5 − n ) = 1 − n . Simplifying, we get − 5 + n = 1 − n . Adding n and 5 to both sides, we get 2 n = 6 , so n = 3 .

Final Answer The equation of line d is y = 3 x − 10 , the value of k is 2 , and the value of n is 3 .


Examples
Understanding parallel lines is crucial in architecture and design. For example, when designing a building, architects use parallel lines to create walls, floors, and ceilings that are aligned and stable. Ensuring that lines are parallel helps maintain structural integrity and aesthetic appeal. In urban planning, parallel streets can optimize traffic flow and create organized city layouts. These concepts ensure efficient use of space and resources.

Answered by GinnyAnswer | 2025-07-07

The equation of the line parallel to y = 3 x − 7 and passing through P ( 4 , 2 ) is y = 3 x − 10 . The value of k such that lines L 1 ​ and L 2 ​ are parallel is 2. The value of n making lines PQ and RS parallel is 3.
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Answered by Anonymous | 2025-07-10