Math Problem Statement

Find k if the line through (5, 11) and (-3, 7) is parallel to the line through (-2, k) and (2, 5).

Solution

To find kk, we need to use the fact that the slopes of two parallel lines must be equal.

Step 1: Calculate the slope of the line passing through (5,11)(5, 11) and (3,7)(-3, 7).

The slope formula between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by:

m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1}

For the points (5,11)(5, 11) and (3,7)(-3, 7):

m1=71135=48=12m_1 = \frac{7 - 11}{-3 - 5} = \frac{-4}{-8} = \frac{1}{2}

So, the slope of the first line is 12\frac{1}{2}.

Step 2: Set the slope of the second line equal to the first, since the lines are parallel.

Now, the slope of the line passing through (2,k)(-2, k) and (2,5)(2, 5) must also be 12\frac{1}{2}. Using the slope formula for these points:

m2=5k2(2)=5k4m_2 = \frac{5 - k}{2 - (-2)} = \frac{5 - k}{4}

Since the slopes must be equal, we set m2m_2 equal to m1m_1:

5k4=12\frac{5 - k}{4} = \frac{1}{2}

Step 3: Solve for kk.

To solve for kk, cross-multiply:

2(5k)=4(1)2(5 - k) = 4(1)

Simplify:

102k=410 - 2k = 4

Now, solve for kk:

104=2k10 - 4 = 2k

6=2k6 = 2k

k=3k = 3

Final Answer:

The value of kk is 33.

Would you like further details or have any questions?


Here are 5 related questions to deepen your understanding:

  1. How do you find the equation of a line given two points?
  2. What happens to the slopes of two perpendicular lines?
  3. How do you find the y-intercept once you have the slope and a point?
  4. How do you determine if three points lie on the same line?
  5. What is the geometric interpretation of a slope of zero?

Tip: For any two points, the slope formula is the ratio of the change in y-coordinates to the change in x-coordinates. This gives a precise measure of how steep a line is!

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Math Problem Analysis

Mathematical Concepts

Algebra
Coordinate Geometry
Slopes of Lines
Parallel Lines

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)

Theorems

Parallel lines have equal slopes

Suitable Grade Level

Grades 8-10