Determine if the lines are parallel, perpendicular, or neither. $\newline$ a) $\newline$ b) $\newline$ \begin{align}y&=-2x+3\-3y&=6x+15\end{align} $\newline$ c) $\newline$ \begin{align}3x+4y&=4\4x+3y&=-21\end{align}

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Q. Determine if the lines are parallel, perpendicular, or neither.

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\begin{align*}y&=-2x+3\-3y&=6x+15\end{align*}

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\begin{align*}3x+4y&=4\4x+3y&=-21\end{align*}

Compare slopes for parallel lines: To determine the relationship between two lines, we need to compare their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular. Otherwise, they are neither parallel nor perpendicular. Let's start with the first pair of equations: $\newline$ $y = -2x + 3$ $\newline$ $-3y = 6x + 15$ $\newline$ First, we need to put the second equation in slope-intercept form, which is $y = mx + b$ , where $m$ is the slope.
Find slope of second line: We will solve the second equation for $y$ to find its slope. $\newline$ $-3y = 6x + 15$ $\newline$ Divide both sides by $-3$ to isolate $y$ : $\newline$ $y = -2x - 5$ $\newline$ Now we have the slope of the second line, which is $-2$ .
Compare slopes for perpendicular lines: Comparing the slopes of the two lines: $\newline$ The slope of the first line is $-2$ , and the slope of the second line is also $-2$ . Since the slopes are equal, the lines are parallel.
Compare slopes for perpendicular lines: Comparing the slopes of the two lines: $\newline$ The slope of the first line is $-2$ , and the slope of the second line is also $-2$ . Since the slopes are equal, the lines are parallel.Now let's move on to the second pair of equations: $\newline$ $3x + 4y = 4$ $\newline$ $4x + 3y = -21$ $\newline$ We need to put both equations in slope-intercept form to compare their slopes.
Compare slopes for perpendicular lines: Comparing the slopes of the two lines: $\newline$ The slope of the first line is $-2$ , and the slope of the second line is also $-2$ . Since the slopes are equal, the lines are parallel.Now let's move on to the second pair of equations: $\newline$ $3x + 4y = 4$ $\newline$ $4x + 3y = -21$ $\newline$ We need to put both equations in slope-intercept form to compare their slopes.First, we solve the third equation for $y$ : $\newline$ $3x + 4y = 4$ $\newline$ Subtract $3x$ from both sides: $\newline$ $4y = -3x + 4$ $\newline$ Divide both sides by $4$ : $\newline$ $y = -\frac{3}{4}x + 1$ $\newline$ The slope of the third line is $-\frac{3}{4}$ .
Compare slopes for perpendicular lines: Comparing the slopes of the two lines: $\newline$ The slope of the first line is $-2$ , and the slope of the second line is also $-2$ . Since the slopes are equal, the lines are parallel.Now let's move on to the second pair of equations: $\newline$ $3x + 4y = 4$ $\newline$ $4x + 3y = -21$ $\newline$ We need to put both equations in slope-intercept form to compare their slopes.First, we solve the third equation for $y$ : $\newline$ $3x + 4y = 4$ $\newline$ Subtract $3x$ from both sides: $\newline$ $4y = -3x + 4$ $\newline$ Divide both sides by $4$ : $\newline$ $y = -\frac{3}{4}x + 1$ $\newline$ The slope of the third line is $-\frac{3}{4}$ .Next, we solve the fourth equation for $y$ : $\newline$ $4x + 3y = -21$ $\newline$ Subtract $4x$ from both sides: $\newline$ $3y = -4x - 21$ $\newline$ Divide both sides by $3$ : $\newline$ $y = -\frac{4}{3}x - 7$ $\newline$ The slope of the fourth line is $-\frac{4}{3}$ .
Compare slopes for perpendicular lines: Comparing the slopes of the two lines: $\newline$ The slope of the first line is $-2$ , and the slope of the second line is also $-2$ . Since the slopes are equal, the lines are parallel.Now let's move on to the second pair of equations: $\newline$ $3x + 4y = 4$ $\newline$ $4x + 3y = -21$ $\newline$ We need to put both equations in slope-intercept form to compare their slopes.First, we solve the third equation for $y$ : $\newline$ $3x + 4y = 4$ $\newline$ Subtract $3x$ from both sides: $\newline$ $4y = -3x + 4$ $\newline$ Divide both sides by $4$ : $\newline$ $y = -\frac{3}{4}x + 1$ $\newline$ The slope of the third line is $-\frac{3}{4}$ .Next, we solve the fourth equation for $y$ : $\newline$ $4x + 3y = -21$ $\newline$ Subtract $4x$ from both sides: $\newline$ $3y = -4x - 21$ $\newline$ Divide both sides by $3$ : $\newline$ $y = -\frac{4}{3}x - 7$ $\newline$ The slope of the fourth line is $-\frac{4}{3}$ .Comparing the slopes of the third and fourth lines: $\newline$ The slope of the third line is $-\frac{3}{4}$ , and the slope of the fourth line is $-\frac{4}{3}$ . These slopes are negative reciprocals of each other, so the lines are perpendicular.