91Ó°ÊÓ

Intercepts are crucial in understanding the position of a line on a graph. They are points where the line touches the axes. To find intercepts of a line equation, one must isolate each variable by setting the other variable to zero.
For example, in the equation of a line, say, \( 3x + 4y = 18 \), you find the x-intercept by setting \( y = 0 \) and solving for \( x \). This gives \( 3x = 18 \), resulting in \( x = 6 \). Similarly, find the y-intercept by setting \( x = 0 \), then \( 4y = 18 \), which simplifies to \( y = \frac{9}{2} \).
These calculations result in two points where the line crosses the axes: the x-intercept (6,0) and y-intercept (0,\( \frac{9}{2} \)).

• Intercepts help to easily draw the line on a graph.
• In our problem, checking the product \( 6 \times \frac{9}{2} = 27 \).
• This specific feature was crucial in identifying the correct equation.

Coordinate geometry is a branch of mathematics that uses algebra to study geometrical shapes. Every point on a plane is defined by a pair of numbers, called coordinates, written as \( (x, y) \). These numbers indicate a point’s exact position.
When dealing with lines, coordinate geometry helps us express the lines with equations, making it easier to analyze their properties and relationships with other lines. A line can often be characterized by two critical components: its slope and intercepts on the axes. In algebra, this understanding allows you to find points of intersection, measure distances, or verify a specific line contains a particular point.
In problems like the one we solved, coordinate geometry allows us to establish if a line truly passes through a known point, like \( (2, 3) \). This point-checking is essential for verifying whether possibilities truly meet all given conditions.

• Serves as a link between geometry and algebra.
• A powerful tool for analyzing linear relations.
• Essential in verifying conditions of specific points and lines.

The equation of a line is a mathematical expression that describes the collection of points that conform to a specific linear pattern on a coordinate plane. Commonly seen as \( y = mx + b \), this form is known as the slope-intercept form, where \( m \) represents the slope and \( b \) the y-intercept.
In the given equation set, the general form \( Ax + By = C \) was used. Transforming either to the slope-intercept form or isolating for intercepts provides different lenses for understanding and solving problems. In equation (3), \( 3x + 4y = 18 \), it clearly showed us the intercepts at \( x = 6 \) and \( y = \frac{9}{2} \). This equation successfully met all conditions required in the task: the product of intercepts equaled 27 and crucially passed through the point \( (2,3) \).
Understanding line equations helps in positioning the line, finding distances between points, and much more.

• Provides the structure to map points on a graph.
• Essential for connecting algebra with graphical representation.
• Versatile form; can be adapted for various applications.