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The slope-intercept form of a linear equation is one of the simplest ways to write and understand linear equations. It is written as: \(\text{y = mx + b}\). In this format, \(m\) represents the slope of the line, and \(b\) represents the y-intercept, which is the point where the line crosses the y-axis. Rewriting equations into this form helps identify key characteristics of the line. For example, starting with an equation like \(3x - y = 12\), we can rearrange it into slope-intercept form by solving for \(y\): 1. Subtract \(3x\) from both sides: \(-y = -3x + 12\)2. Multiply both sides by \(-1\): \(y = 3x - 12\) Now, the equation is in slope-intercept form, with a slope (\(m\)) of 3 and a y-intercept (\(b\)) of -12. This format makes it easier to graph the line and determine its relationship with other lines.

To determine the relationship between two lines, such as whether they are parallel, perpendicular, or neither, you need to examine their slopes. * *Parallel Lines:* Lines are parallel if their slopes (\(m\)) are equal. For example, if one line has a slope of 3 and another also has a slope of 3, the lines are parallel. They will never intersect because they have the same steepness.* *Perpendicular Lines:* Lines are perpendicular if the product of their slopes equals \(-1\). For example, if one line has a slope of 2, a line perpendicular to it would have a slope of \(-1/2\) because \(2 \times -1/2 = -1\). These lines intersect to form a right angle (90 degrees).* *Neither:* If the slopes are neither equal nor their product equals \(-1\), then the lines are neither parallel nor perpendicular.In the exercise, both lines had slopes equal to 3, so they were concluded to be parallel.

A linear equation represents a straight line when graphed on a coordinate plane. The standard form is \(Ax + By = C\), which can be converted into different forms such as slope-intercept form for easier analysis. * *Solving Linear Equations:* To graph a linear equation, you should write it in a form that reveals the slope and y-intercept. Moving to slope-intercept form \(y = mx + b\) is usually a good step. This shows the slope (\(m\)) and the y-intercept (\(b\)) directly.* *Example:* Given the equation \(3x - y = 12\), rearrange it to slope-intercept form to find the slope and intercept:\begin{aligned}& \(3x - y = 12\) \text{ (original equation)} & \( - y = -3x + 12\) \text{ (subtract 3x)} & \( y = 3x - 12\) \text{ (multiply by -1)}ewline* Graphing:* Use the slope and y-intercept to sketch the line. The slope tells you how to move from one point to another on the line (rise over run), while the y-intercept gives you a starting point.* *Multiple Equations:* Compare slopes to determine relationships between lines, such as being parallel or perpendicular, as explained earlier.