{"id":1218,"date":"2025-07-23T20:53:31","date_gmt":"2025-07-23T20:53:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1218"},"modified":"2026-03-20T16:17:56","modified_gmt":"2026-03-20T16:17:56","slug":"systems-of-linear-equations-in-three-variables-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-linear-equations-in-three-variables-learn-it-4\/","title":{"raw":"Systems of Linear Equations in Three Variables: Learn It 4","rendered":"Systems of Linear Equations in Three Variables: Learn It 4"},"content":{"raw":"

Inconsistent Systems of Equations Containing Three Variables<\/h2>\r\nJust as with systems of equations in two variables, we may come across an inconsistent system<\/strong> of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a false statement, such as [latex]3=7[\/latex] or some other contradiction.\r\n\r\n
As you discovered when solving systems that have one solution, well-organized work is essential to being certain about the result you obtain. It can take several steps for the contradiction in a system with no solution to appear, as in the example below.<\/section>
Solve the system.\r\n

[latex]\\begin{align}x - 3y+z=4 && \\left(1\\right) \\\\ -x+2y - 5z=3 && \\left(2\\right) \\\\ 5x - 13y+13z=8 && \\left(3\\right) \\end{align}[\/latex]<\/p>\r\n[reveal-answer q=\"721134\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"721134\"]\r\n\r\nLooking at the coefficients of [latex]x[\/latex], we can see that we can eliminate [latex]x[\/latex] by adding equation (1) to equation (2).\r\n

[latex]\\begin{align}x - 3y+z=4 \\\\ -x+2y - 5z=3 \\\\ \\hline -y - 4z=7\\end{align}[\/latex][latex]\\hspace{5mm} \\begin{align} (1) \\\\ (2) \\\\ (4) \\end{align}[\/latex]<\/p>\r\nNext, we multiply equation (1) by [latex]-5[\/latex] and add it to equation (3).\r\n

[latex]\\begin{align}\u22125x+15y\u22125z&=\u221220 \\\\ 5x\u221213y+13z&=8 \\\\ \\hline 2y+8z&=\u221212\\end{align}[\/latex][latex]\\hspace{5mm} \\begin{align}&(1)\\text{ multiplied by }\u22125 \\\\ &(3) \\\\ &(5) \\end{align}[\/latex]<\/p>\r\nThen, we multiply equation (4) by 2 and add it to equation (5).\r\n

[latex]\\begin{align}\u22122y\u22128z&=14 \\\\ 2y+8z&=\u221212 \\\\ \\hline 0&=2\\end{align}[\/latex][latex]\\hspace{5mm} \\begin{align}&(4)\\text{ multiplied by }2 \\\\ &(5) \\\\& \\end{align}[\/latex]<\/p>\r\nThe final equation [latex]0=2[\/latex] is a contradiction, so we conclude that the system of equations in inconsistent and, therefore, has no solution.\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nIn this system, each plane intersects the other two, but not at the same location. Therefore, the system is inconsistent.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n

Dependent Systems of Equations Containing Three Variables<\/h2>\r\nWe know from working with systems of equations in two variables that a\u00a0dependent system<\/strong>\u00a0of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.\r\n\r\n
Find the solution to the given system of three equations in three variables.\r\n

[latex]\\begin{align}2x+y - 3z=0 && \\left(1\\right)\\\\ 4x+2y - 6z=0 && \\left(2\\right)\\\\ x-y+z=0 && \\left(3\\right)\\end{align}[\/latex]<\/p>\r\n[reveal-answer q=\"633686\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"633686\"]\r\n\r\nFirst, we can multiply equation (1) by [latex]-2[\/latex] and add it to equation (2).\r\n

[latex]\\begin{align} \u22124x\u22122y+6z=0 &\\hspace{9mm} (1)\\text{ multiplied by }\u22122 \\\\ 4x+2y\u22126z=0 &\\hspace{9mm} (2) \\end{align}[\/latex]<\/p>\r\nWe do not need to proceed any further. The result we get is an identity, [latex]0=0[\/latex], which tells us that this system has an infinite number of solutions. There are other ways to begin to solve this system, such as multiplying equation (3) by [latex]-2[\/latex], and adding it to equation (1). We then perform the same steps as above and find the same result, [latex]0=0[\/latex].\r\n\r\nWhen a system is dependent, we can find general expressions for the solutions. Adding equations (1) and (3), we have\r\n

[latex]\\begin{align}2x+y\u22123z=0 \\\\ x\u2212y+z=0 \\\\ \\hline 3x\u22122z=0 \\end{align}[\/latex]<\/p>\r\nWe then solve the resulting equation for [latex]z[\/latex].\r\n

[latex]\\begin{align}3x - 2z=0 \\\\ z=\\frac{3}{2}x \\end{align}[\/latex]<\/p>\r\nWe back-substitute the expression for [latex]z[\/latex] into one of the equations and solve for [latex]y[\/latex].\r\n

[latex]\\begin{align}&2x+y - 3\\left(\\frac{3}{2}x\\right)=0 \\\\ &2x+y-\\frac{9}{2}x=0 \\\\ &y=\\frac{9}{2}x - 2x \\\\ &y=\\frac{5}{2}x \\end{align}[\/latex]<\/p>\r\nSo the general solution is [latex]\\left(x,\\frac{5}{2}x,\\frac{3}{2}x\\right)[\/latex]. In this solution, [latex]x[\/latex] can be any real number. The values of [latex]y[\/latex] and [latex]z[\/latex] are dependent on the value selected for [latex]x[\/latex].\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nAs shown below, two of the planes are the same and they intersect the third plane on a line. The solution set is infinite, as all points along the intersection line will satisfy all three equations.\r\n\r\n\"Two\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

We can express the solution of a dependent system of equations with three variables using only one of the variable.\r\n
    \r\n \t
  • If we choose [latex]x[\/latex] as our variable, you should express [latex]y[\/latex] and [latex]z[\/latex] in term of [latex]x[\/latex] Thus, the general solution would have the formatting: [latex]{(x,y,z)} = {(x, y = mx+b, z = nx+c)}[\/latex].<\/li>\r\n \t
  • If we choose [latex]z[\/latex] as our variable, the general solution would be [latex]{(x,y,z)} = {(x = mz+a, y = nz+b, z)}[\/latex].<\/li>\r\n<\/ul>\r\n<\/section>
    [ohm_question hide_question_numbers=1]321622[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]321623[\/ohm_question]<\/section>","rendered":"

    Inconsistent Systems of Equations Containing Three Variables<\/h2>\n

    Just as with systems of equations in two variables, we may come across an inconsistent system<\/strong> of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a false statement, such as [latex]3=7[\/latex] or some other contradiction.<\/p>\n

    As you discovered when solving systems that have one solution, well-organized work is essential to being certain about the result you obtain. It can take several steps for the contradiction in a system with no solution to appear, as in the example below.<\/section>\n
    Solve the system.<\/p>\n

    [latex]\\begin{align}x - 3y+z=4 && \\left(1\\right) \\\\ -x+2y - 5z=3 && \\left(2\\right) \\\\ 5x - 13y+13z=8 && \\left(3\\right) \\end{align}[\/latex]<\/p>\n