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A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in different forms, but one of the most common is the slope-intercept form, which is expressed as

\( y = mx + b \)

where \( m \) represents the slope of the line, and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis. This form is particularly useful because it allows you to quickly identify both the steepness of the line and where it intercepts the y-axis.
In our original exercise, the goal was to rearrange the given equation into slope-intercept form. Doing this helps clarify how the line represented by the equation behaves when graphed on a coordinate plane.

The ability to manipulate algebraic expressions and solve for variables is crucial in algebra. The process involves several key techniques such as distributing, combining like terms, and isolating the variable. In the exercise given,

\( y - 30.0 = 0.265(x - 10) \)

the first step involves distributing the \( 0.265 \) to both terms within the parentheses. It's equivalent to multiplying \( 0.265 \) by both \( x \) and \( -10 \), which simplifies the equation. The next step is to isolate \( y \) by performing the inverse operation (adding \( 30 \)) to both sides of the equation, which cancels out the negative term. These manipulations simplify the equation, making it easy to identify the slope and y-intercept of the linear equation.

Graphing linear equations is a way of visually representing the solutions of the equation. To graph a linear equation in slope-intercept form,

\( y = mx + b \),

we first plot the y-intercept \( b \) on the y-axis. From there, we use the slope \( m \), which tells us the rise over run, to find another point on the line. For example, a slope of \( 0.265 \) means that for each unit increase in \( x \), the value of \( y \) increases by \( 0.265 \). By plotting multiple points using the slope and drawing a straight line through them, the linear equation is graphed. The slope becomes the angle of the line relative to the horizontal axis, and the y-intercept is where the line crosses the y-axis. Understanding how to visually represent linear equations can immensely help in quickly grasping their properties and how they relate to various applications in mathematics.