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Coordinate geometry is a branch of mathematics that combines algebra and geometry. It allows us to use algebraic methods to solve geometric problems. This is achieved by representing figures and shapes through pairs of numbers.

In coordinate geometry, a point is identified by two coordinates: the x-coordinate and the y-coordinate. These coordinates define the position of the point on a plane. For example, the point (6, 2) is located 6 units along the x-axis and 2 units up along the y-axis.

• Coordinate plane: A two-dimensional plane where each point is determined by a pair of numerical values.
• X-axis and Y-axis: The horizontal and vertical lines used to determine positions on the coordinate plane.
• Ordered pairs: A pair of numbers, such as (6, 2), used to indicate the location of a point.

Coordinate geometry helps us understand the relationship between algebraic equations and geometric figures, such as lines and curves.

Linear equations are equations of the first degree, meaning they consist of variables raised only to the power of one. In the context of coordinate geometry, a linear equation describes a straight line on the coordinate plane.

In the equation \[-3x + 4y = -5\]This represents a line where \(-3\) and \(4\) are the coefficients of \(x\) and \(y\) respectively, and \(-5\) is the constant term.

• Slope-intercept form: A common form of the linear equation is \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
• Coefficients: Numbers that multiply the variables (e.g., \(-3\) and \(4\) in our equation).
• Constant term: The constant part of the equation, which affects the position of the line but not its slope.

Linear equations are essential for graphing straight lines and analyzing their behavior.

The distance formula is a valuable tool in coordinate geometry for calculating the distance between a point and a line, as well as between two points. To find the distance from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\), we use the formula: \[\text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}.\]
The formula works by measuring the perpendicular distance from the point to the line, ensuring it is the shortest distance possible. For instance, in the given problem, we substitute the values of \(A = -3\), \(B = 4\), \(C = -5\), \(x_1 = 6\), and \(y_1 = 2\) into the formula and calculate:

• Calculate the absolute value of \(-3(6) + 4(2) - 5 = |-8| = 8\).
• Find the square root of \((-3)^2 + 4^2 = \sqrt{25} = 5\).
• Divide 8 by 5 to get the distance \(1.6\).

Understanding the distance formula helps us determine how close a point is to a given line, providing insight into spatial relationships on the coordinate plane.