In Exercises 67-76, write the slope-intercept forms of the...

In Exercises 67-76, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line.
$\left(-\frac{2}{3}, \frac{7}{8}\right), \quad 3 x+4 y=7$

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Key Concepts

Slope-Intercept Form

The slope-intercept form is an equation of a line written as y = mx + b, where m is the slope and b is the y-intercept. This form is particularly useful for quickly identifying the steepness of the line and the point at which it crosses the y-axis.

Slope

The slope is a measure of the steepness and direction of a line, defined as the ratio of the vertical change to the horizontal change between any two points on the line. It is essential for interpreting and constructing linear equations from various forms.

Parallel Lines

Parallel lines are lines in the same plane that never intersect, meaning they have identical slopes. This characteristic is used to ensure that a new line will run in the same direction as a given line.

Perpendicular Lines

Perpendicular lines intersect to form right angles (90 degrees). The slopes of perpendicular lines are negative reciprocals of each other, a relationship that is crucial when determining the equation of a line perpendicular to a given line.

Transcript

00:01 So we have this equation for a line.

00:02 I'm going to get this equation into slope intercept form first.

00:06 So we're going to have four y is equal to negative 3x plus 7.

00:09 I just minus 3x to both sides.

00:12 Now we can divide both sides by 4.

00:13 So we have y is equal to negative 3 4ths.

00:17 X plus 7 4ths.

00:19 And so the first thing we're going to do is find an equation for a line that's parallel to this line.

00:24 This is 3 4ths.

00:25 I'm going to make sure that doesn't look like negative 3 divided by y.

00:29 And so in order for a line to be parallel, we need their slopes to be equal.

00:34 So our slope here is going to be negative three -fourths.

00:37 So we're going to have y is equal to negative three -fourths times x plus b.

00:42 And we're going to try and solve for this b value.

00:45 So the point that we want this line to go through is the point negative two -thirds, comma, seven -eighths.

00:53 So we're going to plug those points in for x and y and figure out what b is equal to.

00:57 So we're going to have 7 eighths is equal to negative two -thirds times negative three -fourths plus b.

01:09 So negative two -thirds times negative two -fourths is going to be equal to one -half...

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