Problem 32 - Algebra 1: Concepts and Skills

Question

Write in standard form an equation of the line that passes through the two points. Use integer coefficients. \((0,0),(2,0)\)

Step-by-Step Solution

Answer

The equation of the line in standard form is \(y = 0\).

Step 1: Find the slope

The slope of a line through two points (x1,y1) and (x2,y2) is calculated using the formula: \(M = (y2 - y1)/(x2 - x1)\). However, here you can see from the given points, (0,0) and (2,0), the two y-values are equal. So, the slope \(M = 0\). The line is horizontal.

Step 2: Write the Equation in Slope-Intercept Form

The slope-intercept form of the line is \(y = Mx + b\), where M is the slope and b is the y-intercept. The y-intercept is the y-coordinate of the point where the line crosses the y-axis. Since M = 0 and the line passes through the origin (0,0), the y-intercept is 0. So, the equation of the line is \(y = 0\). This is already in slope-intercept form.

Step 3: Convert to Standard Form

The standard form of a linear equation is \(Ax + By = C\), where A, B, and C are integers, and A and B are not both zero. Since the equation of our line is \(y = 0\), we can convert it to standard form by multiplying all terms by 1 to keep A, B and C as integers. This gives \(0x + 1y = 0\), and simplifying leads to the final standard form equation: \(y = 0\).

Key Concepts

Slope

The slope of a line is a measure of its steepness. It tells us how much the y-coordinate of a point on the line changes for a unit change in the x-coordinate. In mathematics, slope is often denoted by the letter \(M\).

To calculate the slope between two points, you can use 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 the two points.

For example, if you have two points \((0,0)\) and \((2,0)\), plug them into the formula:

• \( M = \frac{0 - 0}{2 - 0} = 0 \)

Thus, the slope is zero, indicating that the line is horizontal.
In real life, a slope of zero indicates no incline, similar to a flat surface.

Standard Form of a Line

The standard form of a line is a specific way to write the equation of a line. It simplifies understanding by bringing constants and variables to one side of the equation. It is written as:

• \( Ax + By = C \)

Where \( A, B, \) and \( C \) are integers, and \( A \) and \( B \) are not both zero.

In the exercise given, the line passes through the origin and is horizontal, with an equation \( y = 0 \).

To convert it into standard form, apply the formula:

• Multiply the equation by an integer to keep coefficients as integers.
• This gives \( 0x + 1y = 0 \), simplifying to \( y = 0 \).

The simplicity of converting from slope-intercept to standard form is especially visible with horizontal lines.

Coordinate Geometry

Coordinate geometry is a field of mathematics that uses algebra to study geometric properties. Specifically, it investigates figures using the coordinate system. A key aspect is the understanding of lines using points plotted in a coordinate plane.

In coordinate geometry, each point is identified by an ordered pair \((x, y)\). These pairs describe locations on a plane.

The equation of a line through two points provides information about the line's direction and position. For instance, with points \((0,0)\) and \((2,0)\), the line remains on the x-axis because both points share the same y-coordinate.

By plotting these and similar points, one can visualize how properties like slope or form affect the line’s appearance. Thus, coordinate geometry serves as a tool to bridge numerical understanding with spatial visualization.