Problem 109 - College Algebra

Question

Find the equation of the line that passes through the following points: \((a, 0)\) and \((c, d)\).

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{d}{c-a}x - \frac{da}{c-a} \).

Step 1: Understand the Slope Formula

The slope (m) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] \Here our points are \((a, 0)\) and \((c, d)\).

Step 2: Substitute the Points into the Slope Formula

Substitute the points \((a, 0)\) and \((c, d)\) into the slope formula. This gives: \\[ m = \frac{d - 0}{c - a} = \frac{d}{c-a} \] \This is the slope of the line.

Step 3: Use the Point-Slope Form of a Line Equation

The point-slope form of the equation of a line is: \\[ y - y_1 = m(x - x_1) \] \where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Using point \((a, 0)\) and slope \(m = \frac{d}{c-a}\), the equation becomes: \\[ y - 0 = \frac{d}{c-a}(x - a) \] which simplifies to \[ y = \frac{d}{c-a}(x - a) \]

Step 4: Simplify the Equation If Necessary

Distribute the slope to simplify the equation: \\[ y = \frac{d}{c-a}x - \frac{da}{c-a} \] \This equation represents the line passing through the points \((a, 0)\) and \((c, d)\).

Key Concepts

Slope Formula

The slope formula is a fundamental concept in coordinate geometry. It helps you find the tilt or steepness of a line connecting two distinct points in a plane. To calculate the slope (\(m \), you subtract the y-coordinate of the first point (\(y_1 \) from the y-coordinate of the second point (\(y_2 \). Then, divide this result by the difference in the x-coordinates of the two points (\(x_2 - x_1 \).

• The formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
• The slope is a measure of how much the line rises or falls as you move from left to right.

If the slope is positive, the line rises; if it is negative, the line falls. A slope of zero means the line is horizontal, indicating no vertical change as you move along the line. Using slope is a simple way to understand the direction and steepness of a line.

Point-Slope Form

Understanding the point-slope form is key when you are given a point on a line and the slope. This form allows you to write the equation of a line efficiently. The point-slope form is expressed as:

• \[ y - y_1 = m(x - x_1) \]
• This shows that if you have a known point \((x_1, y_1)\) and a slope \(m\), there's an easy way to write the equation of the line.

For instance, if a line passes through point \((a, 0)\) with slope \( \frac{d}{c-a} \), you establish the line equation as:\[ y - 0 = \frac{d}{c-a}(x - a) \], simplifying to:\[ y = \frac{d}{c-a}(x - a) \].This format helps in visualizing lines in coordinate geometry because it clearly shows where the line starts and how it progresses based on its slope.

Line Equation

The line equation is a central theme in both algebra and geometry. It describes the relationship between x and y coordinates on a Cartesian plane. The general form of a line equation is \( y = mx + b \), where \(m\) is the slope, and \(b\) is the y-intercept. However, there are different forms depending on the information given, like the point-slope form.When simplifying equations for lines, you ensure that all points along the line satisfy the given relationship. For example, a simplified line equation derived from point \((a, 0)\) and \((c, d)\) is \[ y = \frac{d}{c-a}x - \frac{da}{c-a} \]. This form shows how x and y connect through linear relationships, expressing proportionality and transformation.

Coordinate Geometry

Coordinate geometry integrates algebra and geometry to visualize and analyze shapes using a coordinate plane. By plotting points (combinations of x and y) in this plane, you can effectively study lines, curves, and shapes.Using coordinate geometry, you identify a line by plotting points and using formulas like the slope and point-slope form. This branch of mathematics is crucial for solving real-world problems because it provides a visual representation.

• By locating points like \((a, 0)\) and \((c, d)\), you efficiently calculate the slope and derive line equations.
• These equations allow precise descriptions of placements and movements on a plane.

Coordinate geometry forms the basis for more advanced studies in calculus and helps in practical fields like engineering, computer graphics, and physics.