91Ó°ÊÓ

Coordinate geometry, also known as analytical geometry, is a field in mathematics that uses algebraic equations to describe geometric figures and relationships. It's all about locating points on the Cartesian plane using ordered pairs
, typically denoted as \((x, y)\). These coordinates describe the position of a point in a 2D space relative to the origin, \((0, 0)\).
In an ordered pair, the first number is the horizontal position, labeled as \(x\), while the second number is the vertical position, labeled as \(y\).
To understand how linear equations fit into coordinate geometry, think of the equation \(y = mx + b\), known as the slope-intercept form:

• The slope \(m\) determines the tilts of the line.
• The intercept \(b\) is where the line crosses the \(y\)-axis.

This form helps to easily plot a straight line on a plane. Each solution \((x, y)\) of the equation represents a point on this line.

Solving equations involves finding the value(s) of unknown variables that make the equation true. In linear equations, these unknowns typically appear as \(x\) or \(y\) and are solved to find their specific values.
To solve an equation means to rearrange terms to isolate the variable of interest. For example, given \(y = -5x + 3\):

• First, identify which variable you need to solve for, based on the context, usually by substituting known values.
• Substitute known values into the equation.
• Simplify the equation to find the value of the unknown variable.

This method ensures that you can determine specific points \((x, y)\) that satisfy the linear equation.

The substitution method is a powerful tool used especially when dealing with systems of equations. However, it is also applicable when verifying if a point lies on a line.
In context, it means replacing one variable with a known value to solve for the other. Consider our situation where we analyze whether the point \((-1, k)\) belongs to the line \(y = -5x + 3\):

• Substitute \(x = -1\) into the equation, turning it into \(k = -5(-1) + 3\).
• Calculate to find \(k\). This gives you \(k = 8\).

By substituting the value of \(x\), we simplify the problem to a single-variable equation, making it easier to solve and confirm that a point lies on a given line.