Characterization of Parallel Lines

Characterization of Parallel Lines exercise

• Determine if the given pairs of lines are intersecting or parallel.

  Hints

  Slope-intercept form for a line is $y=mx+b$. The following lines are in slope-intercept form:

  $\begin{array}{l} y=x+10\\ y=\frac12 x-7\\ y=-5x+\frac23\\ y=6\\ \end{array}$

  Line with the same slope are parallel. Their $y$-intercepts must be different, otherwise they would be the same line. The following lines are parallel:

  $\begin{array}{l} y=\frac56 x+3\\ y=\frac56 x\\ y=\frac56 x - 4\\ \end{array}$

  Standard form for a line is $ax+by=c$. Here are some examples:

  $\begin{array}{rcc} -3x+4y &=& 9\\ \frac67 x-9y &=& 0\\ x-y &=& 4\\ \end{array}$

  We can convert a line from standard form to slope-intercept form by solving for $y$. Here's an example:

  $\begin{array}{rcl} 2x+3y&=&6\\ \mathbf{-2x}&&\mathbf{-2x}\\ \\ 3y&=&-2x+6\\ \mathbf{\div3}&&\mathbf{\div3}\\ \\ y&=&-\frac23 x +2\\ \end{array}$

  Solution

  Here is the first pair of lines:

  $\begin{array}{l} y=2x+1\\ y=5x-8\\ \end{array}$

  • These lines are in slope-intercept form ($y=mx+b$).
  • Their slopes are $m=2$ and $m=5$, so their slopes are different.
  • That means the lines must intersect.
  • To solve the system, we create the equation:

  $2x+1=5x-8$

  • Solving this gives us $x=3$, which means the system has a solution.
  • This confirms that the lines must intersect.

  Here is the second pair of lines:

  $\begin{array}{l} -x+y=-3\\ -x+y=2\\ \end{array}$

  • These lines are in standard form ($ax+by=c$).
  • We convert them to slope-intercept form by solving each for $y$:

  $\begin{array}{l} y=x-3\\ y=x+2\\ \end{array}$

  • The slope of each line is $m=1$, so the slopes are the same.
  • Lines with the same slope are parallel--they don't intersect.
  • To solve the system, we create the equation:

  $x-3=x+2$

  • Solving this gives us $-3=2$, which means the system has a solution.
  • This is a false statement, so the system has no solution.
  • This confirms that the lines do not intersect.

• Visualize the graphs of parallel lines.

  Hints

  Slope-intercept form for a line is $y=mx+b$.

  Given the following equations for lines, we can find the slope and y-intercept, as shown in the table.

  \begin{array}{l|c|c|c} \text{equation} & \text{slope} & \text{y-intercept} & \text{point on line}\\ \hline &&&\\ y=-\frac15 x-1 & -\frac15 & -1 & (0,-1)\\ &&&\\ y=3x+7 & 3 & 7 & (0,7) \end{array}

  The line $y=\frac23 x + 8$ is in slope-intercept form. To graph it:

  • Notice that the y-intercept is $8$.
  • Graph the point $(0,8)$.
  • Notice that the slope is $\frac23$.
  • From the y-intercept, we rise $2$ and run $3$, to arrive at the point (3,10).
  • Graph the line through $(0,8)$ and $(3,10)$.

  • A positive slope means the line slopes up from left to right.
  • A negative slope means the line slopes down from left to right.

  Given a line which is not in slope-intercept form, solve for $y$ in order to convert it into slope-intercept form. Here's an example:

  $\begin{array}{lll} 2x+3y&=&6\\ \mathbf{-2x}&&\mathbf{-2x}\\ \\ 3y&=&-2x+6\\ \mathbf{\div3}&&\mathbf{\div3}\\ \\ y&=&-\frac23 x +2\\ \end{array}$

  Solution

  $\begin{array}{l|l|c|c|c} \text{equation} & \text{slope-intercept form}& \text{slope} & \text{y-intercept} & \text{point on line}\\ \hline &&&\\ y=2x & y=2x & 2 & 0 & (0,0)\\ y=2x+3 & y=2x+3 & 2 & 3 & (0,3)\\ &&&\\ x=y+5 & y=x-5 & 1 & -5 &(0,-5)\\ x=y-1 & y=x+1 & 1 & 1 & (0,1)\\ &&&\\ y=-\frac12 x + 3 & y=-\frac12 x+3 & -\frac12 & 3 &(0,3)\\ y=-\frac12 x + 2 & y=-\frac12 x+2 & -\frac12 & 2 &(0,2)\\ &&&\\ 3y-x=0 & y=\frac13 x & \frac13 & 0 & (0,0)\\ x+3y=-12 & y=\frac13 x-4 & \frac13 & 4 & (0,4)\\ &&&\\ y=x+3 & y=x+3 & 1 & 3 & (0,3)\\ y=x+2 & y=x+2 & 1 & 2 & (0,2)\\ \end{array}$

• Determine if the two equations represent parallel or intersecting lines.

  Hints

  • If lines are in slope-intercept form, we can compare their slopes and y-intercepts to determine if the lines are parallel.

  • If the slopes are the same and the $y$-intercepts are different, the lines are parallel.

  • If the slopes are different, the lines are intersecting.

  • Both lines in the pair below are in slope-intercept form.

  $\begin{array}{l} y=2x-6\\ y=3x+10\\ \end{array}$

  • The slope of the first line is $2$.

  • The slope of the second line is $3$.

  • Because the slopes are different, the lines must intersect.

  • If lines are in slope-intercept form, we can solve the system of equations to determine if there is an intersection or not.

  • Given the lines below:

  \begin{array}{l} y=2x-2\\ y=6\\ \end{array}

  • We can solve this system by setting $2x-2$ equal to $6$, and solving.

  $\begin{array}{l} 2x-2=6\\ 2x=8\\ x=4 \end{array}$

  • Because we got a solution for $x$, we know the lines intersect.

  The following lines are not in slope-intercept form:

  \begin{array}{l} 3x=y-9\\ x+2y=3\\ \end{array}

  • Isolate $y$ in each to convert them to slope-intercept form:

  \begin{array}{l|l} \text{first line} & \text{second line}\\ \hline 3x=y-9 & x+2y=3\\ 3x+9=y & 2y=-x+3\\ y=3x+9 & y=\frac{-x+3}{2}\\ & y=-\frac12 x+\frac32 \end{array}

  • The first line has a slope of $3$ and the second has a slope of $-\frac12$.

  • These lines have different slopes, so they must intersect.

  • We can confirm this by solving the system:

  $\begin{array}{l} 3x+9 = -\frac12 x+\frac32\\ 6x+18=-x+3\\ 7x=-15\\ x=-\frac{15}{7} \end{array}$

  • Because solving the system gave us a solution, we know the lines are not parallel.

  Solution

  ___________________

  $\begin{array}{l} y=4x-7\\ y=4x+8\\ \end{array}$

  • The above lines have the same slope, $4$, and different $y$-intercepts of $-7$ and $8$, so they are parallel.

  ___________________

  $\begin{array}{l} y=6x+2\\ x=6y+2\\ \end{array}$

  • Re-arranging the above lines so they are in slope-intercept form gives us:

  $\begin{array}{l} y=6x+2\\ y=\frac16 x - \frac13\\ \end{array}$

  • The above lines have different slopes, so they are intersecting.

  ___________________

  $\begin{array}{l} y=x-7\\ x=y\\ \end{array}$

  • Re-arranging the above lines so they are in slope-intercept form gives us:

  $\begin{array}{l} y=x-7\\ y=x\\ \end{array}$

  The lines have the same slope, $1$, and different $y$-intercepts, which means they are parallel.

  ___________________

  $\begin{array}{l} x+2y=10\\ x+2y=20\\ \end{array}$

  • You might be able to tell immediately that when you put the above lines in slope-intercept form, they will have the same slope and different $y$-intercepts. They are parallel:

  $\begin{array}{l} y=-\frac12 x + 5\\ y=-\frac12 x + 10\\ \end{array}$

  ___________________

  $\begin{array}{l} 2x+5y=30\\ 2x+5y=40\\ \end{array}$

  • You might be able to tell immediately that when you put the above lines in slope-intercept form, they will have the same slope and different $y$-intercepts. They are parallel:

  $\begin{array}{l} y=-\frac25 x + 6\\ y=-\frac25 x + 8\\ \end{array}$

  ___________________

  $\begin{array}{l} 2x+5y=30\\ 2x-5y=-30\\ \end{array}$

  • In slope-intercept form:

  $\begin{array}{l} y=-\frac25 x + 6\\ y=\frac25 x + 6\\ \end{array}$

  • The slopes are different, so the lines intersect.

  ___________________

  $\begin{array}{l} 3x=y+5\\ 3x=y+9\\ \end{array}$

  • In slope-intercept form:

  $\begin{array}{l} y=3x-5\\ y=3x-9\\ \end{array}$

  ___________________

  $\begin{array}{l} x=y\\ x=-y\\ \end{array}$

  • In slope-intercept form:

  $\begin{array}{l} y=x\\ y=-x\\ \end{array}$

  • The slopes are different, so the lines intersect.

  ___________________

  $\begin{array}{l} y=7\\ y=x+7\\ \end{array}$

  • Both lines are in slope-intercept form, $y=mx+b$.
  • The first line can be written as $y=0x+7$.
  • That means the slope of the first line is $0$.
  • The slope of the second line is $1$.
  • The slopes are different, so the lines intersect.

• Determine which lines are parallel to each other.

  Hints

  • If two lines are in slope-intercept form, compare their slopes and $y$-intercepts to determine if they are parallel.

  • Given the following pair of lines in slope-intercept form:

  \begin{array}{l} y=-\frac25 x +6\\ y=\frac25 x + 7 \end{array}

  • The slopes are different, so the lines are not parallel.

  • If two lines are in slope-intercept form, compare their slopes and $y$-intercepts to determine if they are parallel.

  • Given the following pair of lines in slope-intercept form:

  \begin{array}{l} y=7x +6\\ y=7x + 7 \end{array}

  • The slopes are the same and the $y$-intercepts are different, so the lines are parallel.

  The following lines are not in slope-intercept form:

  \begin{array}{l} 3x=y-9\\ x+2y=3\\ \end{array}

  • Isolate $y$ in each to convert them to slope-intercept form:

  \begin{array}{l|l} \text{first line} & \text{second line}\\ \hline 3x=y-9 & x+2y=3\\ 3x+9=y & 2y=-x+3\\ y=3x+9 & y=\frac{-x+3}{2}\\ & y=-\frac12 x+\frac32 \end{array}

  • The first line has a slope of $3$ and the second has a slope of $-\frac12$.

  • These lines have different slopes, so they must intersect.

  • We can confirm this by solving the system:

  $\begin{array}{l} 3x+9 = -\frac12 x+\frac32\\ 6x+18=-x+3\\ 7x=-15\\ x=-\frac{15}{7} \end{array}$

  • Because solving the system gave us a solution, we know the lines are not parallel.

  • We can also directly compare lines which are not in slope-intercept form.

  Consider these two lines:

  $\begin{array}{l} 5x+8y=11\\ 5x+8y=12\\ \end{array}$

  • Both lines are in standard form, that is, $ax+by=c$.

  • Without isolating $y$ in each we can already tell that the result will be the same slope and different $y$-intercepts.

  • This is because the coefficients of $x$ and $y$ are the same, namely $5$ and $8$, and the constant terms, $11$ and $12$, are different.

  • Let's directly compare two more lines which are not in slope-intercept form.

  Consider these two lines:

  $\begin{array}{l} 5x+8y=11\\ -5x-8y=11\\ \end{array}$

  • Both lines are in standard form, that is, $ax+by=c$.

  • Let's multiply both sides of the second equation by $-1$ to get:

  $\begin{array}{l} 5x+8y=11\\ 5x+8y=-11\\ \end{array}$

  • Without isolating $y$ in each we can already tell that the result will be the same slope and different y-intercepts.

  • This is because the coefficients of $x$ and $y$ are the same, namely $5$ and $8$, and the constant terms, $11$ and $-11$, are different.

  Solution

  Parallel Lines:

  \begin{array}{l} y=5x-11\\ y+12=5x\\ \end{array}

  • Converting to slope-intercept form gives us:

  \begin{array}{l} y=5x-11\\ y=5x-12\\ \end{array}

  • These lines have the same slope, $5$, and different $y$-intercepts, $-11$ and $-12$.

  • Therefore, these two lines are parallel.

  Parallel Lines:

  \begin{array}{l} x+y=4\\ x+y=0\\ \end{array}

  • Both of these lines are in standard form. The coefficients of $x$ and $y$ are the same, and the constant terms are different.

  • Therefore, these two lines are parallel.

  • Isolating $y$ to convert them to slope-intercept form will confirm this.

  Parallel Lines:

  \begin{array}{l} 3x+4y=9\\ y=-\frac34 x -2 \end{array}

  • Converting the first line to slope-intercept form gives us:

  \begin{array}{l} y=-\frac34 x + \frac94\\ y=-\frac34 x -2 \end{array}

  • The slopes are the same and the $y$-intercepts are different, so the lines are parallel.

  \begin{array}{l} x=\frac14 y +1\\ x+1=\frac14 y\\ \end{array}

  • Converting both of these lines to standard form gives us:

  \begin{array}{l} x-\frac14 y =1\\ x-\frac14 y=-1\\ \end{array}

  • The coefficients of $x$ and $y$ are the same for both lines, and the constant terms are different.

  • This means the lines are parallel.

  • Isolating $y$ in each to convert each to slope-intercept form confirms this.

• Graph the line corresponding to the equation $y=-\frac34x+3$.

  Hints

  • Given a line in the form $y=mx+b$, the y-intercept is $b$.

  • For example, given the line $y=-\frac57 x + 9$, the $y$-intercept is $9$.

  • That means the line goes through the point $(0,9)$.

  • The y-intercept is the $y$-value corresponding to an $x$-value of $0$.

  • It's the point where the graph crosses the $y$-axis.

  • The $x$-intercept is the $x$-value corresponding to a $y$-value of $0$.

  • It's the point where the graph crosses the $x$-axis.

  • We can find the $y$-intercept for a linear equation by substituting $0$ for $x$ and finding $y$.

  • For example, given the equation $y=6x-7$:

  $\begin{array}{l} y=6(0)+7\\ y=7\\ \end{array}$

  • The $y$-intercept is $7$.

  • That means the graph goes through the point $(0,7)$.

  • We can find the x-intercept for a linear equation by substituting $0$ for $y$ and finding $x$.

  • For example, given the equation $y=6x-7$:

  $\begin{array}{l} 0=6x+7\\ 6x=-7\\ x=-\frac76\\ \end{array}$

  • The $y$-intercept is $-\frac76$.

  • That means the graph goes through the point $(-\frac76,0)$.

  Solution

  The equation is $y=-\frac34 x + 3$

  The $y$-intercept

  • The equation is in slope-intercept form.
  • The $y$-intercept is $3$.
  • That means the graph goes through the point $(0,3)$.

  The $x$-intercept

  • We find an $x$-intercept by substituting $0$ for $y$ and solving for $x$:

  $\begin{array}{l|l} 0=-\frac34 x+ 3& \text{Subtract 3 from both sides.}\\ &\\ -3=-\frac34 x & \text{Multiply both sides by} -\frac43\\ &\\ (-\frac43)(-3)=(-\frac43)(-\frac34 x) &\text{Cancel and reduce.}\\ &\\ 4=x&\\ \end{array}$

  • The $x$-intercept is $4$.
  • That means the graph goes through the point $(4,0)$.

  The line through $(0,3)$ and $(4,0)$ is the graph of $y=-\frac34 x +3$.

• Determine how many solutions each pair of equations has.

  Hints

  • Lines that have different slopes and the same or different y-intercepts are intersecting.

  • These lines intersect:

  $\begin{array}{l} y=6x\\ y=12x+7 \end{array}$

  • These lines also intersect:

  $\begin{array}{l} y=3x+9\\ y=4x+9 \end{array}$

  • The above lines happen to intersect at their common y-intercept, the point $(0,9)$.

  • Two lines that have different slopes intersect in one point.

  • Lines that have the same slope and different y-intercepts are parallel.

  • These lines are parallel:

  $\begin{array}{l} y=9x\\ y=9x+8 \end{array}$

  • Because the two lines are parallel, they have no intersection point.

  • The fact that the y-intercepts are different means these are two distinct lines.

  • If the y-intercepts were the same, we would have:

  $\begin{array}{l} y=9x+8\\ y=9x+8 \end{array}$

  • That would mean these two lines were in fact the same line.

  • If two lines are in fact the same line, they share all of their points, which means there are infinitely many intersection points.

  • Lines that have the same slope and the same y-intercept are coincident.

  • These lines are coincident:

  $\begin{array}{l} y=10x-3\\ 2y=20x-6 \end{array}$

  • When we convert the second line to slope-intercept form, we get:

  $\begin{array}{l} y=10x-3\\ y=10x-3 \end{array}$

  • These two lines are in fact the same line.

  • If two lines are in fact the same line, they share all of their points, which means there are infinitely many intersection points.

  Solution

  $\begin{array}{l} y=2x+8\\ y=x+8 \end{array}$

  • Different slope, same y-intercept.
  • The fact that the y-intercept is the same is irrelevant.
  • They intersect at one point.
  • Solving $2x+8=x+8$ gives us one solution.

  $\begin{array}{l} y=5x-4\\ y=5x-8 \end{array}$

  • Same slope, different y-intercept.
  • They are parallel.
  • They have no intersection point.
  • Solving $5x-4=5x-8$ gives us a false statement.
  • The fact that the y-intercepts are different means we do in fact have two distinct lines.

  $\begin{array}{l} y=7x+3\\ 2y=14x+6 \end{array}$

  • Converting to slope-intercept form gives us:

  $\begin{array}{l} y=7x+3\\ y=7x+3 \end{array}$

  • Same slope, same y-intercept.
  • They are coincident.
  • They intersect in an infinite number of points.
  • Solving $7x+3=7x+3$ gives us $0=0$, a statement which is true for all values of $x$.