91Ó°ÊÓ

The substitution method is a handy technique used to solve a system of equations or verify if a given point satisfies an equation. The idea is to replace one part of the equation with a known value to simplify the process. In the provided exercise, we had the equation \( y = -6x + 1 \) and needed to check if the point \((9, k)\) is a solution. To do this, we substituted \( x = 9 \) into the equation to find the missing \( y \)-value, which in this case is expressed as \( k \).
Here's a quick breakdown of how substitution simplifies our problem:

• Replace the variable in the equation with the provided value.
• Simplify the mathematical expressions by following basic arithmetic operations.
• Reach a solution, or determine the missing value in an equation.

This method is commonly used because it's straightforward and only requires basic algebra skills for solving linear equations.

Coordinate geometry, also known as analytic geometry, allows us to use algebraic methods to solve geometric problems. When we talk about a point like \((9, k)\) in this context, we're referring to coordinates on the Cartesian plane where \( 9 \) is the \( x \)-coordinate and \( k \) is the \( y \)-coordinate.
Each point on this plane can be described uniquely by these two values:

• \( x \)-coordinate: Represents the horizontal position on the plane.
• \( y \)-coordinate: Represents the vertical position on the plane.

The coordinate geometry makes it possible to visualize these values on the axis, helping us to see how equations like \( y = -6x + 1 \) form lines on the graph. In our exercise, understanding that \( (9, k) \) refers to a specific location on this plane is key to finding \( k \) when the line is defined by our linear equation. Such visual and algebraic interpretations are crucial in mathematics.

Solving an equation typically means finding the values of its variable that make the equation true. When we have a linear equation in the form \( y = -6x + 1 \), solving it involves determining \( y \) for a given \( x \).
In our case, the goal was to find \( k \) so the point \((9, k)\) fits on the line described by the equation. We substituted \( x = 9 \) into the equation:
\[ y = -6(9) + 1 \]
Simplifying gives:

• Calculate \(-6 \times 9\) to get \(-54 \).
• Add 1 to \(-54\) which results in \(-53\).

This solving process confirmed \( y = -53 \), telling us that \( k = -53 \). Such solutions help in determining particular points that lie along the line represented by the equation, thus demonstrating the real-world application of linear equations.