Question

Are two lines, one passing through \((6,9)\) and \((2,4)\) and another with slope \(\frac{5}{4}\) and passing through the point \((2,1)\), parallel?

Short answer

Yes, the lines are parallel because they have equal slopes \(\frac{5}{4}\).

Step-by-step solution

Calculate the slope of the first line

Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) to find the slope of the line passing through the points \(6,9\) and \(2,4\). Substitute the coordinates: \[m = \frac{4 - 9}{2 - 6} = \frac{-5}{-4} = \frac{5}{4}\].

Identify the slope of the second line

The slope of the second line is given directly as \(\frac{5}{4}\).

Compare the slopes

Two lines are parallel if and only if their slopes are equal. Compare the slope of the first line (calculated in Step 1) with the slope of the second line (given in Step 2). Both slopes are \(\frac{5}{4}\).

Conclusion

Since the slopes of the two lines are equal \(\frac{5}{4} = \frac{5}{4}\), the two lines are parallel.

Key concepts

Slope Calculation

To determine if two lines are parallel, we need to understand how to calculate their slopes.
Slope is a measure of how steep a line is and can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two different points on the line.
For example, to find the slope of a line passing through \( (6, 9) \) and \( (2, 4) \), we substitute these values into the formula:
\[ m = \frac{4 - 9}{2 - 6} = \frac{-5}{-4} = \frac{5}{4} \]
Remember, simplifying negative fractions can be tricky, but \( - \div - = + \), so the slope \( m = \frac{5}{4} \).
Identifying the slope correctly is critical as it helps determine the line's direction and steepness.

Coordinate Geometry

Coordinate geometry deals with the positions of points on a plane using their numerical coordinates.
To work with lines in coordinate geometry, you first need to understand the basics:

• The coordinate plane is divided into four quadrants.
• Every point on this plane is identified by an ordered pair (x, y).
• The x-value indicates the horizontal position, and the y-value indicates the vertical position.

Lines on this plane can be represented using equations like y = mx + b, where m is the slope and b is the y-intercept.
Using coordinate geometry, we can calculate the distance between points, the slope of lines, and check if lines are parallel or perpendicular.
For instance, in our exercise, we used points (6, 9) and (2, 4) to find the line's slope and determined if it is parallel to the line with slope \( \frac{5}{4} \) that passes through (2, 1).

GED Math Problems

GED (General Educational Development) math problems often cover a range of topics including slope calculation and coordinate geometry.
To excel in these problems, you need to:

• Understand core concepts such as slope and line equations.
• Be proficient at substituting values into formulas and simplifying results.
• Know how to interpret and solve problems involving points on a coordinate plane.
  

Solving the given problem involves all these steps - calculating slope, using coordinates, and applying the concept of parallel lines.
Start by carefully reading the problem to identify what is given and what needs to be calculated.
In this case, we calculated the slopes and determined that the slopes are equal, hence the lines are parallel.
Practice similar problems frequently to build confidence and accuracy in solving GED math problems.