{"id":13198,"date":"2015-11-24T18:11:12","date_gmt":"2015-11-24T18:11:12","guid":{"rendered":"https:\/\/courses.candelalearning.com\/precalcone1xcleanmaster\/?post_type=chapter&p=13198"},"modified":"2015-11-24T18:11:12","modified_gmt":"2015-11-24T18:11:12","slug":"identifying-and-expressing-solutions-to-systems-of-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/identifying-and-expressing-solutions-to-systems-of-equations\/","title":{"raw":"Identifying and Expressing Solutions to Systems of Equations","rendered":"Identifying and Expressing Solutions to Systems of Equations"},"content":{"raw":"

Now that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an inconsistent system<\/strong> consists of parallel lines that have the same slope but different [latex]y[\/latex] -intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as [latex]12=0[\/latex].\n<\/p>

\n

Example 8: Solving an Inconsistent System of Equations<\/h3>\nSolve the following system of equations.\n
[latex]\\begin{array}{l}\\text{ }x=9 - 2y\\hfill \\\\ x+2y=13\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n
\n

Solution<\/h3>\nWe can approach this problem in two ways. Because one equation is already solved for [latex]x[\/latex], the most obvious step is to use substitution.\n
[latex]\\begin{array}{l}x+2y=13\\hfill \\\\ \\left(9 - 2y\\right)+2y=13\\hfill \\\\ 9+0y=13\\hfill \\\\ 9=13\\hfill \\end{array}[\/latex]<\/div>\nClearly, this statement is a contradiction because [latex]9\\ne 13[\/latex]. Therefore, the system has no solution.\n\nThe second approach would be to first manipulate the equations so that they are both in slope-intercept form. We manipulate the first equation as follows.\n
[latex]\\begin{array}{l}\\text{ }x=9 - 2y\\hfill \\\\ 2y=-x+9\\hfill \\\\ \\text{ }y=-\\frac{1}{2}x+\\frac{9}{2}\\hfill \\end{array}[\/latex]<\/div>\nWe then convert the second equation expressed to slope-intercept form.\n
[latex]\\begin{array}{l}x+2y=13\\hfill \\\\ \\text{ }2y=-x+13\\hfill \\\\ \\text{ }y=-\\frac{1}{2}x+\\frac{13}{2}\\hfill \\end{array}[\/latex]<\/div>\nComparing the equations, we see that they have the same slope but different y<\/em>-intercepts. Therefore, the lines are parallel and do not intersect.\n
[latex]\\begin{array}{l}\\begin{array}{l}\\\\ y=-\\frac{1}{2}x+\\frac{9}{2}\\end{array}\\hfill \\\\ y=-\\frac{1}{2}x+\\frac{13}{2}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n
\n

Analysis of the Solution<\/h3>\nWriting the equations in slope-intercept form confirms that the system is inconsistent because all lines will intersect eventually unless they are parallel. Parallel lines will never intersect; thus, the two lines have no points in common. The graphs of the equations in this example are shown in Figure 8.\n\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]\"AFigure 8<\/b>[\/caption]\n\n<\/div>\n
\n

Try It 6<\/h3>\nSolve the following system of equations in two variables.\n
[latex]\\begin{array}{l}2y - 2x=2\\\\ 2y - 2x=6\\end{array}[\/latex]<\/div>\nSolution<\/a>\n\n<\/div>\n
\n

Expressing the Solution of a System of Dependent Equations Containing Two Variables<\/h2>\nRecall that a dependent system<\/strong> of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or addition, the resulting equation will be an identity, such as [latex]0=0[\/latex].\n
\n

Example 9: Finding a Solution to a Dependent System of Linear Equations<\/h3>\nFind a solution to the system of equations using the addition method<\/strong>.\n
[latex]\\begin{array}{c}x+3y=2\\\\ 3x+9y=6\\end{array}[\/latex]<\/div>\n<\/div>\n
\n

Solution<\/h3>\nWith the addition method, we want to eliminate one of the variables by adding the equations. In this case, let\u2019s focus on eliminating [latex]x[\/latex]. If we multiply both sides of the first equation by [latex]-3[\/latex], then we will be able to eliminate the [latex]x[\/latex] -variable.\n
[latex]\\begin{array}{l}\\text{ }x+3y=2\\hfill \\\\ \\left(-3\\right)\\left(x+3y\\right)=\\left(-3\\right)\\left(2\\right)\\hfill \\\\ \\text{ }-3x - 9y=-6\\hfill \\end{array}[\/latex]<\/div>\nNow add the equations.\n

[latex]\\begin{array} \\hfill\u22123x\u22129y=\u22126 \\\\ \\hfill+3x+9y=6 \\\\ \\hfill \\text{_____________} \\\\ \\hfill 0=0 \\end{array}[\/latex]<\/p>\nWe can see that there will be an infinite number of solutions that satisfy both equations.\n\n<\/div>\n

\n

Analysis of the Solution<\/h3>\nIf we rewrote both equations in the slope-intercept form, we might know what the solution would look like before adding. Let\u2019s look at what happens when we convert the system to slope-intercept form.\n
[latex]\\begin{array}{l}\\text{ }x+3y=2\\hfill \\\\ \\text{ }3y=-x+2\\hfill \\\\ \\text{ }y=-\\frac{1}{3}x+\\frac{2}{3}\\hfill \\\\ 3x+9y=6\\hfill \\\\ \\text{ }9y=-3x+6\\hfill \\\\ \\text{ }y=-\\frac{3}{9}x+\\frac{6}{9}\\hfill \\\\ \\text{ }y=-\\frac{1}{3}x+\\frac{2}{3}\\hfill \\end{array}[\/latex]<\/div>\nSee Figure 9. Notice the results are the same. The general solution to the system is [latex]\\left(x, -\\frac{1}{3}x+\\frac{2}{3}\\right)[\/latex].\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"AFigure 9<\/b>[\/caption]\n\n<\/div>\n
\n

Try It 7<\/h3>\nSolve the following system of equations in two variables.\n
[latex]\\begin{array}{l}\\begin{array}{l}\\\\ \\text{ }\\text{}\\text{}y - 2x=5\\end{array}\\hfill \\\\ -3y+6x=-15\\hfill \\end{array}[\/latex]<\/div>\n
Solution<\/a><\/div>\n<\/div>\n<\/div>","rendered":"

Now that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an inconsistent system<\/strong> consists of parallel lines that have the same slope but different [latex]y[\/latex] -intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as [latex]12=0[\/latex].\n<\/p>\n

\n

Example 8: Solving an Inconsistent System of Equations<\/h3>\n

Solve the following system of equations.<\/p>\n

[latex]\\begin{array}{l}\\text{ }x=9 - 2y\\hfill \\\\ x+2y=13\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n
\n

Solution<\/h3>\n

We can approach this problem in two ways. Because one equation is already solved for [latex]x[\/latex], the most obvious step is to use substitution.<\/p>\n

[latex]\\begin{array}{l}x+2y=13\\hfill \\\\ \\left(9 - 2y\\right)+2y=13\\hfill \\\\ 9+0y=13\\hfill \\\\ 9=13\\hfill \\end{array}[\/latex]<\/div>\n

Clearly, this statement is a contradiction because [latex]9\\ne 13[\/latex]. Therefore, the system has no solution.<\/p>\n

The second approach would be to first manipulate the equations so that they are both in slope-intercept form. We manipulate the first equation as follows.<\/p>\n

[latex]\\begin{array}{l}\\text{ }x=9 - 2y\\hfill \\\\ 2y=-x+9\\hfill \\\\ \\text{ }y=-\\frac{1}{2}x+\\frac{9}{2}\\hfill \\end{array}[\/latex]<\/div>\n

We then convert the second equation expressed to slope-intercept form.<\/p>\n

[latex]\\begin{array}{l}x+2y=13\\hfill \\\\ \\text{ }2y=-x+13\\hfill \\\\ \\text{ }y=-\\frac{1}{2}x+\\frac{13}{2}\\hfill \\end{array}[\/latex]<\/div>\n

Comparing the equations, we see that they have the same slope but different y<\/em>-intercepts. Therefore, the lines are parallel and do not intersect.<\/p>\n

[latex]\\begin{array}{l}\\begin{array}{l}\\\\ y=-\\frac{1}{2}x+\\frac{9}{2}\\end{array}\\hfill \\\\ y=-\\frac{1}{2}x+\\frac{13}{2}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n
\n

Analysis of the Solution<\/h3>\n

Writing the equations in slope-intercept form confirms that the system is inconsistent because all lines will intersect eventually unless they are parallel. Parallel lines will never intersect; thus, the two lines have no points in common. The graphs of the equations in this example are shown in Figure 8.<\/p>\n

\"A<\/p>\n

Figure 8<\/b><\/p>\n<\/div>\n<\/div>\n

\n

Try It 6<\/h3>\n

Solve the following system of equations in two variables.<\/p>\n

[latex]\\begin{array}{l}2y - 2x=2\\\\ 2y - 2x=6\\end{array}[\/latex]<\/div>\n

Solution<\/a><\/p>\n<\/div>\n

\n

Expressing the Solution of a System of Dependent Equations Containing Two Variables<\/h2>\n

Recall that a dependent system<\/strong> of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or addition, the resulting equation will be an identity, such as [latex]0=0[\/latex].<\/p>\n

\n

Example 9: Finding a Solution to a Dependent System of Linear Equations<\/h3>\n

Find a solution to the system of equations using the addition method<\/strong>.<\/p>\n

[latex]\\begin{array}{c}x+3y=2\\\\ 3x+9y=6\\end{array}[\/latex]<\/div>\n<\/div>\n
\n

Solution<\/h3>\n

With the addition method, we want to eliminate one of the variables by adding the equations. In this case, let\u2019s focus on eliminating [latex]x[\/latex]. If we multiply both sides of the first equation by [latex]-3[\/latex], then we will be able to eliminate the [latex]x[\/latex] -variable.<\/p>\n

[latex]\\begin{array}{l}\\text{ }x+3y=2\\hfill \\\\ \\left(-3\\right)\\left(x+3y\\right)=\\left(-3\\right)\\left(2\\right)\\hfill \\\\ \\text{ }-3x - 9y=-6\\hfill \\end{array}[\/latex]<\/div>\n

Now add the equations.<\/p>\n

[latex]\\begin{array} \\hfill\u22123x\u22129y=\u22126 \\\\ \\hfill+3x+9y=6 \\\\ \\hfill \\text{_____________} \\\\ \\hfill 0=0 \\end{array}[\/latex]<\/p>\n

We can see that there will be an infinite number of solutions that satisfy both equations.<\/p>\n<\/div>\n

\n

Analysis of the Solution<\/h3>\n

If we rewrote both equations in the slope-intercept form, we might know what the solution would look like before adding. Let\u2019s look at what happens when we convert the system to slope-intercept form.<\/p>\n

[latex]\\begin{array}{l}\\text{ }x+3y=2\\hfill \\\\ \\text{ }3y=-x+2\\hfill \\\\ \\text{ }y=-\\frac{1}{3}x+\\frac{2}{3}\\hfill \\\\ 3x+9y=6\\hfill \\\\ \\text{ }9y=-3x+6\\hfill \\\\ \\text{ }y=-\\frac{3}{9}x+\\frac{6}{9}\\hfill \\\\ \\text{ }y=-\\frac{1}{3}x+\\frac{2}{3}\\hfill \\end{array}[\/latex]<\/div>\n

See Figure 9. Notice the results are the same. The general solution to the system is [latex]\\left(x, -\\frac{1}{3}x+\\frac{2}{3}\\right)[\/latex].<\/p>\n

\"A<\/p>\n

Figure 9<\/b><\/p>\n<\/div>\n<\/div>\n

\n

Try It 7<\/h3>\n

Solve the following system of equations in two variables.<\/p>\n

[latex]\\begin{array}{l}\\begin{array}{l}\\\\ \\text{ }\\text{}\\text{}y - 2x=5\\end{array}\\hfill \\\\ -3y+6x=-15\\hfill \\end{array}[\/latex]<\/div>\n
Solution<\/a><\/div>\n<\/div>\n<\/div>\n\n\t\t\t
\n\t\t\t

Candela Citations<\/h3>\n\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
CC licensed content, Specific attribution<\/div>