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One method of solving a system of equations is by graphing. We know that the graph of a linear equation in two variables is a straight line. The graph of a system will consist of two straight lines. When two straight lines are graphed, one of three possibilities may result.
The lines intersect at the point
$\left(a,\text{\hspace{0.17em}}b\right)$ . The point
$\left(a,\text{\hspace{0.17em}}b\right)$ is the solution to the corresponding system.
The lines are parallel. They do not intersect. The system has no solution.
The lines are coincident (one on the other). They intersect at infinitely many points. The system has infinitely many solutions.
Solve each of the following systems by graphing.
$\{\begin{array}{ccc}\begin{array}{r}2x+y=5\\ x+y=2\end{array}& & \begin{array}{l}\left(1\right)\\ \left(2\right)\end{array}\end{array}$
Write each equation in slope-intercept form.
$\begin{array}{lllllll}\left(1\right)\hfill & \hfill & -2x+y=5\hfill & \hfill & \left(2\right)\hfill & \hfill & x+y=2\hfill \\ \hfill & \hfill & y=2x+5\hfill & \hfill & \hfill & \hfill & y=-x+2\hfill \end{array}$
Graph each of these equations.
The lines appear to intersect at the point
$\left(-1,\text{\hspace{0.17em}}3\right)$ . The solution to this system is
$\left(-1,\text{\hspace{0.17em}}3\right)$ , or
$\begin{array}{ccc}x=-1,& & y=3\end{array}$
$\begin{array}{llllllllllll}(1)\hfill & \hfill & \hfill -2x+y& =\hfill & 5\hfill & \hfill & (2)\hfill & \hfill & \hfill x+y& =\hfill & 2\hfill & \hfill \\ \hfill & \hfill & \hfill -2(-1)+3& =\hfill & 5\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill & \hfill & \hfill -1+3& =\hfill & 2\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 2+3& =\hfill & 5\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill & \hfill & \hfill 2& =\hfill & 2\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill \\ \hfill & \hfill & \hfill 5& =\hfill & 5\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \end{array}$
$\{\begin{array}{ccc}\begin{array}{l}-x+y=-1\\ -x+y=2\end{array}& & \begin{array}{l}\left(1\right)\\ \left(2\right)\end{array}\end{array}$
Write each equation in slope-intercept form.
$\begin{array}{rrrrrrrrrrr}\hfill \left(1\right)& \hfill & \hfill -x+y& \hfill =& -1\hfill & \hfill & \hfill \left(2\right)& \hfill & \hfill -x+y& \hfill =& 2\hfill \\ \hfill & \hfill & \hfill y& \hfill =& x-1\hfill & \hfill & \hfill & \hfill & \hfill y& \hfill =& x+2\hfill \end{array}$
Graph each of these equations.
These lines are parallel. This system has no solution. We denote this fact by writing
inconsistent .
We are sure that these lines are parallel because we notice that they have the same slope,
$m=1$ for both lines. The lines are not coincident because the
$y$ -intercepts are different.
$\{\begin{array}{ccc}\begin{array}{l}-2x+3y=-2\\ -6x+9y=-6\end{array}& & \begin{array}{l}\left(1\right)\\ \left(2\right)\end{array}\end{array}$
Write each equation in slope-intercept form.
$\begin{array}{rrrrrrrrrrr}\hfill \left(1\right)& \hfill & \hfill -2x+3y& \hfill =& -2\hfill & \hfill & \hfill \left(2\right)& \hfill & \hfill -6x+9y& \hfill =& -6\hfill \\ \hfill & \hfill & \hfill 3y& \hfill =& 2x-2\hfill & \hfill & \hfill & \hfill & \hfill 9y& \hfill =& 6x-6\hfill \\ \hfill & \hfill & \hfill y& \hfill =& \hfill \frac{2}{3}x-\frac{2}{3}& \hfill & \hfill & \hfill & \hfill y& \hfill =& \hfill \frac{2}{3}x-\frac{2}{3}\end{array}$
Both equations are the same. This system has infinitely many solutions. We write
dependent .
Solve each of the following systems by graphing. Write the ordered pair solution or state that the system is inconsistent, or dependent.
$\{\begin{array}{l}-2x+3y=6\\ 6x-9y=-18\end{array}$
dependent
$\{\begin{array}{l}3x+5y=15\\ 9x+15y=15\end{array}$
inconsistent
For the following problems, solve the systems by graphing. Write the ordered pair solution, or state that the system is inconsistent or dependent.
$\{\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x+y=-5\\ -x+y=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\end{array}$
$\left(-3,-2\right)$
$\{\begin{array}{l}x+y=4\\ x+y=0\end{array}$
$\{\begin{array}{l}-3x+y=5\\ -\text{\hspace{0.17em}}\text{\hspace{0.17em}}x+y=3\end{array}$
$\left(-1,2\right)$
$\{\begin{array}{l}x-y\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=-6\\ x\text{\hspace{0.17em}}+2y=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\end{array}$
$\{\begin{array}{l}3x+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=0\\ 4x-3y=\text{\hspace{0.17em}}12\end{array}$
$\left(\frac{12}{13},-\frac{36}{13}\right)$
$\{\begin{array}{l}-4x+\text{\hspace{0.17em}}y=7\\ -3x+\text{\hspace{0.17em}}y=2\end{array}$
$\{\begin{array}{l}2x+3y=\text{\hspace{0.17em}}6\\ 3x+4y=\text{\hspace{0.17em}}6\end{array}$
These coordinates are hard to estimate. This problem illustrates that the graphical method is not always the most accurate.
$\left(-6,6\right)$
$\{\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=-3\\ 4x+\text{\hspace{0.17em}}4y=-12\end{array}$
$\{\begin{array}{l}2x-3y=1\\ 4x-6y=4\end{array}$
inconsistent
$\{\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x+2y=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3\\ -3x-6y=-9\end{array}$
$\{\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x-2y=\text{\hspace{0.17em}}\text{\hspace{0.17em}}6\\ 3x-6y=18\end{array}$
dependent
$\{\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2x\text{\hspace{0.17em}}+3y=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}6\\ -10x-15y=30\end{array}$
( [link] ) Express $0.000426$ in scientific notation.
$4.26\times {10}^{-4}$
( [link] ) Find the product: ${\left(7x-3\right)}^{2}.$
(
[link] ) Supply the missing word. The
slope
(
[link] ) Supply the missing word. An equation of the form
$a{x}^{2}+bx+c=0,a\ne 0$ , is called a
(
[link] ) Construct the graph of the quadratic equation
$y={x}^{2}-3.$
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