Parallel and Perpendicular Lines

Consider the given equation of the given line. Parallel lines have the same slope. Therefore, the slope of all lines parallel to the given line is $2.$ A general equation in slope-intercept form for these lines can be written. The desired parallel line passes through $(\text{-}2,3).$ By substituting the coordinates of this point into the above equation for $x$ and $y,$ the $y\text{-}$intercept $b$ of the parallel line can be determined. Knowing the $y\text{-}$intercept, the equation of the line parallel to $y=2x-1$ through $(\text{-}2,3)$ can be written. The solution can be verified by graphing the line on the same coordinate plane.

As seen above, the graph of $y=2x+7$ passes through $(\text{-}2,3)$ and is parallel to the graph of $y=2x-1.$

Two parallel lines have the same slope. To find the equation of the line that corresponds to the bike path, the slope of the line that represents the pavement has to be determined first. This can be done by using the Slope Formula. In the above formula, $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line. Therefore, two points on the line corresponding to the pavement have to be identified. Now $(0,3)$ and $(1,1)$ will be substituted for $(x_1,y_1)$ and $(x_2,y_2),$ respectively. The slope of the pavement is $\text{-}2.$ As mentioned before, parallel lines have the same slope. Therefore, all lines parallel to the line that corresponds to the pavement will have a slope of $\text{-}2.$ A general equation in slope-intercept form for these lines can be written. The line that represents the bike path passes through $(\text{-}4,3.5).$ By substituting the coordinates of this point into the above equation for $x$ and $y,$ the $y\text{-}$intercept $b$ of the parallel line can be determined. Knowing the $y\text{-}$intercept, the equation of the line parallel to the pavement through $(\text{-}4,3.5)$ can be written. The solution will be verified by plotting the line on Kevin's graph.

As seen above, the graph of $y=\text{-}2x-4.5$ is parallel to the pavement and passes through $(\text{-}4,3.5).$

The coordinates of the points that represent the stars are $(\text{-}4,\text{-}3)$ and $(3,0.5).$ Since two points on the line are known, the line's slope can be found using the Slope Formula. The slope of the line that passes through the stars is $\frac{1}{2},$ or $0.5.$ Parallel lines have the same slope. Therefore, all lines that are parallel to the line that passes through the stars will have a slope of $\frac{1}{2}.$ A general equation in slope-intercept form for these lines can be written. The parallel line should pass through the moon, which in the coordinate plane has coordinates $(\text{-}2,1).$ These two numbers will be substituted into the above equation. Then, the $y\text{-}$intercept $b$ of the parallel line can be determined. Now, the equation of the parallel line can be completed. The solution will be verified by graphing the above line on the visual representation of Zosia's proposed mural.

Consider the given equation. Two lines are perpendicular if and only if their slopes are negative reciprocals. This means that their product is $\text{-}1.$ The slope of the given equation is $\text{-}3.$ By substituting this value into the above equation the slope of a perpendicular line $m_2$ can be found. All perpendicular lines to the line whose equation is given have a slope of $\frac{1}{3}.$ A general equation in slope-intercept form for these lines can be written. The desired perpendicular line passes through the point with coordinates $(6,\text{-}1).$ By substituting these numbers into the obtained equation for $x$ and $y,$ the $y\text{-}$intercept $b$ of the perpendicular line can be determined. Knowing the $y\text{-}$intercept, the equation of the perpendicular line to $y=\text{-}3x+2$ through $(6,\text{-}1)$ can now be written. The solution can be verified by graphing this line on the same coordinate plane.

As seen above, the graph of $y=\frac{1}{3}x-3$ is perpendicular to the graph of $y=\text{-}3x+2$ and passes through $(6,\text{-}1).$

First, the slope of the line that represents the main pipe will be determined. To do so, the Slope Formula will be used. Here, $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of two points on the line. Therefore, two points on the given line are needed. The coordinates of two points on the main pipe are $(0,2)$ and $(2,3).$ These will be substituted into the formula. The slope of the line that represents the main pipe is $\frac{1}{2}.$ Two lines are perpendicular when their slopes are negative reciprocals. This means that the product of the slopes is $\text{-}1.$ The slope of the given line is $\frac{1}{2}.$ By substituting this value into the above equation for $m_1,$ the slope of a perpendicular line $m_2$ can be found. All lines perpendicular to the line that represents the main pipe have a slope of $\text{-}2.$ A general equation in slope-intercept form for these lines can be written. The line that corresponds to the new pipe should pass through $(3,\text{-}1).$ By substituting $3$ and $\text{-}1$ in the general equation for $x$ and $y,$ the $y\text{-}$intercept $b$ of the perpendicular line can be determined. Knowing the $y\text{-}$intercept, the equation of the new pipe can now be written. The solution can be verified by graphing the line on the given plan.

As seen above, the graph of $y=\text{-}2x+5$ is perpendicular to the given line and passes through $(3,\text{-}1).$ The new pipe is a part of $y=\text{-}2x+5.$

The stone path that connects the bench with the entrance lies on the line that passes through the points with coordinates $(0,4)$ and $(6,0).$ These can be used to find the line's slope by using the Slope Formula. The $x\text{-}$coordinate of $(0,4)$ is $0.$ Therefore, the $y\text{-}$intercept of the line through this point is $4.$ Knowing the slope an the $y\text{-}$intercept of the line, its equation in slope-intercept form can be written. The path from the bench to the entrance is a part of the line described by the above equation. Note that the whole equation is not actually needed. It would be enough to know only the slope of the line passing through the bench and the entrance to find a line perpendicular to it. Now, the equation of the line that corresponds to the second path must be found. First, the slope of this line will be determined. The second line is perpendicular to $y=\text{-}\frac{2}{3}x+4.$ Perpendicular lines have negative reciprocal slopes. This means that their product is $\text{-}1.$ The slope of the known equation is $\text{-}\frac{2}{3}.$ By substituting this value into the above equation for $m_1,$ the slope of a perpendicular line $m_2$ can be found. All lines perpendicular to the line that passes through the bench and the entrance have a slope of $\frac{3}{2}.$ A general equation in slope-intercept form for these lines can be written. The desired line has to allow community members to easily access the flower beds. This means that the line should pass through the point with coordinates $(6,4).$ By substituting them into the above equation for $x$ and $y,$ the $y\text{-}$intercept $b$ of the perpendicular line can be found. Knowing the $y\text{-}$intercept, the equation of the line perpendicular to $y=\text{-}\frac{2}{3}x+4$ through $(6,4)$ can be written. The path that will let visitors to the garden access the flowers is a part of the above line. Below it is shown how the solution looks in the specialized software.

It can be seen that the lines are perpendicular and that $y=\frac{3}{2}x-5$ passes through $(6,4),$ which corresponds to the flower beds.