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The slope-intercept form is a way of writing the equation of a straight line. It is given by the formula:\[ y = mx + b \]Here, 'm' is the slope of the line, and 'b' is the y-intercept, the point where the line crosses the y-axis. Converting an equation to this form helps you easily identify the slope and y-intercept.For example, the equation \(2x + 5y = -7\) can be rearranged to find 'y' by isolating it on one side:1. Subtract '2x' from both sides: \[ 5y = -2x - 7 \]2. Divide every term by 5: \[ y = \frac{-2}{5}x - \frac{7}{5} \]Now the equation is in slope-intercept form, where the slope \(m = \frac{-2}{5}\). This method helps in quickly determining the characteristics of the line in question.

Negative reciprocals are pairs of numbers that, when multiplied together, yield -1. This concept is crucial in determining if two lines are perpendicular.For instance, if you have a slope \(m_1\), the slope of a line that is perpendicular to this line will be \( -\frac{1}{m_1} \).Looking at our example from earlier: 1. The slope of the first line is \( \frac{-2}{5} \).2. The slope of the second line is \( \frac{5}{2} \).We check if they are negative reciprocals by multiplying them together: \( \frac{-2}{5} \times \frac{5}{2} = -1 \).Since the product is -1, the lines are perpendicular. Recognizing negative reciprocals helps us easily determine the relationship between two lines.

Understanding line slopes is fundamental for different aspects of geometry and algebra. The slope of a line measures its steepness and direction, and it's calculated as the rise over the run (change in y over change in x).The slope-intercept form mentioned above gives a straightforward way to find the slope. For example, in the line equation \(y = mx + b\), 'm' is the slope.Slope criteria help us identify whether lines are parallel, perpendicular, or neither:

• Parallel lines have equal slopes (\( m_1 = m_2 \)).
• Perpendicular lines have slopes that are negative reciprocals (\( m_1 = -\frac{1}{m_2} \)).
• Lines that do not meet either criterion are neither parallel nor perpendicular.
Understanding this allows us to categorize the relationships of lines effectively. Applying this to our example, recognizing that one line's slope is \( \frac{-2}{5} \) and the other's is \( \frac{5}{2} \), we determined they are perpendicular due to their slopes being negative reciprocals of each other.