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A linear equation is a mathematical expression that represents a straight line on a graph. It is typically written in the standard form, \( y = mx + b \), where

• \( y \) is the dependent variable, or the output.
• \( x \) is the independent variable, or the input.
• \( m \) is the slope of the line, indicating its steepness and direction.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Linear equations are fundamental in coordinate geometry because they help us understand relationships between variables. For instance, the equation \( y = 2x + 3 \) shows that for every unit increase in \( x \), \( y \) increases by 2, starting from 3 on the y-axis. The line always grants a unique solution for \( y \) for any given \( x \). Most problems in coordinate geometry involve finding such connections or determining specific values that satisfy these equations.

Finding a solution to an equation means identifying the values of variables that make the equation true. When you have a point's coordinates, like \((-9, k)\), you can place these into a linear equation to see if they produce a valid equality.

• For example, when substituting \( x = -9 \) into the equation \( y = 2x + 3 \), you calculate \( y = 2(-9) + 3 = -15 \).
• In this case, \( y \) equals \(-15 \), confirming that \( k = -15 \) in the point \((-9, k)\).

In coordinate geometry, ensuring a point is a solution can help verify if it lies on a given line. It is a way to provide evidence that particular coordinates fulfill the equation set by the line. Such solutions reveal not just specific points but the broader pattern or path dictated by the equation.

The substitution method is a useful technique in solving systems of equations or verifying solutions. It involves replacing a variable with a numerical or another algebraic expression. This approach simplifies the equation, allowing us to solve for an unknown variable. For example:

• Start with the equation \( y = 2x + 3 \).
• Substitute a known value for \( x \), such as \(-9\), to find \( y \).
• Calculate \( y = 2(-9) + 3 \), simplifying it step-by-step to find \( y = -15 \).

By applying the substitution method, we efficiently find that \( k \) must be equal to \(-15\) for the point \((-9, k)\) to satisfy the given linear equation. This method is both practical and intuitive, providing clarity in solving or checking any two-variable equation for correctness. The key to success with substitution is careful arithmetic and methodical steps, ensuring every substitution is accurate and straightforward.