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A linear equation represents a mathematical statement where two expressions are set equal to each other, and the highest power of the variable is one. They are essential because they model relationships between two quantities in many practical problems, like predicting income based on hours worked or the distance traveled over time.
One common form of a linear equation is the slope-intercept form, written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. However, not all linear equations start in this format. They often need rearrangement, just like our example \( \frac{1}{2}x - 6y = -3 \).
Linear equations can be visually represented with a straight line on a graph. Key characteristics include:

• Constant rate of change, meaning the slope (\(m\)) remains consistent throughout.
• One solution, if it intersects a given horizontal line at one point, indicating the value of \(y\) for any \(x\).

Solving for a variable in an equation involves manipulating the equation to express one variable solely in terms of another. This isolates the desired variable, making the equation much clearer and easier to interpret.
In our original equation \( \frac{1}{2} x - 6 y = -3 \), we solved for \( y \) by isolating it through a series of steps:
1. **Eliminate fractions**: Multiply the entire equation by 2 to get rid of the fraction. This leads to \( x - 12y = -6 \).
2. **Rearrange terms**: Add \( 12y \) to both sides to move it to the other side of the equation, giving \( x + 6 = 12y \).
3. **Isolate the variable**: Finally, divide each side of the equation by 12, resulting in \( y = \frac{x + 6}{12} \).
This process of solving for \( y \) transforms the original equation, allowing us to better understand how \( y \) changes with \( x \). Factors like the coefficient's size and the sign (positive or negative) impact the nature of the relationship.

A function in mathematics is a rule that assigns each input exactly one output. In simple terms, if you input a value, you will always get a specific result associated with that value.
A common way to determine if an equation defines a function is to check if each \( x \) value corresponds to only one \( y \) outcome. For our expression \( y = \frac{x+6}{12} \), as \( x \) changes, it calculates only one \( y \). This confirms it as a function of \( x \).
Key traits of functions include:

• **Unique mapping**: Each input has one and only one output.
• **Graph appearance**: A function will pass the vertical line test. If no vertical line intersects the graph at more than one point, it confirms it is a function.

Linear equations, such as the solved form of \( y = \frac{x+6}{12} \), typically represent functions due to this direct input-output relationship.