91Ó°ÊÓ

In geometry, a circle equation represents a circle's collection of points in a coordinate plane. The standard form of a circle's equation is

• \((x-h)^2 + (y-k)^2 = r^2\)

where

• \((h, k)\) is the center of the circle
• \(r\) is the radius.

However, the equation \(x^2 + y^2 - x - 3y = 0\) given in the exercise is not in standard form. By completing the square, we rearrange this to compare with the standard form. Completing the square involves reorganizing terms:

• \((x - \frac{1}{2})^2 + (y - \frac{3}{2})^2 = \frac{5}{4}\)

Thus, the circle's center is \((\frac{1}{2}, \frac{3}{2})\) and the radius is \(\sqrt{\frac{5}{4}}\). By understanding the circle's geometric position, solving intersection problems become more intuitive.

A line equation often represents a straight line on a coordinate plane. The simplest form of a line equation is the slope-intercept form:

• \(y = mx + c\)

Here's what it tells us:

• \(m\) is the slope of the line, indicating its steepness. In the given equation \(y = x - 1\), \(m = 1\), meaning a slope of 1 indicates a positive 45-degree inclination.
• \(c\) is the y-intercept, the point where the line crosses the y-axis. Here, \(c = -1\), meaning the line crosses the y-axis at -1.

This form makes it easy to graph and understand the behavior of the line. For finding intersection points, we substitute this equation into others.

Quadratic equations come in the standard form

• \(ax^2 + bx + c = 0\)

where

• \(a\), \(b\), and \(c\) are constants
• \(aeq 0\).

In our exercise, after substitution and simplification from the circle equation, we obtain

• \(x^2 - 3x + 2 = 0\)

This is a quadratic equation. Such equations can be solved by factoring, completing the square, or using the quadratic formula. Here, factoring was applied, resulting in

• \((x - 1)(x - 2) = 0\)

This provided roots \(x = 1\) and \(x = 2\). These solutions direct us toward potential intersection points with the line.

To solve equations involving systems of a circle and line, a precise process is pivotal. First, recognize and note each equation prominently. Start by substituting and simplifying:

• Replace one variable from a simpler equation into the complex equation.

Always follow up with careful expansion and combination of like terms to form a quadratic equation from the circle equation substituting in the line's equation.

• Ensure you simplify as much as possible before solving.

Proceed to find solutions using factorization, providing you insights or solutions to each variable. With each solution of \(x\), use the line equation to calculate corresponding \(y\)-values. Confirm these points as intersections by checking if they satisfy both original equations. Clearly recapping with final points of intersection ensures comprehensive understanding.