## Systems of Linear Equations in Three Variables

 3.1 Systems of Linear Equations in Three Variables

 Definition Two or more linear equations involving the same set of variables form a system of linear equations.

 General Form of a Linear Equation in Three Variables The general form of a linear equation in three variables can be written as follows: $$ax+by+cz=d$$, where $$a$$, $$b$$, and $$c$$ are not equal to zero.

Example of Systems of Linear Equations in Three Variables
 System of Equations Description \begin{aligned} 2x+4y-z&=10\\ x+y&=10z^2\\ 5y-z-2x&=3 \end{aligned} Not systems of linear equations, because there is an equation in which the highest power of the variable is $$2$$. \begin{aligned} p+8q-4r&=2\\ 2(p+6r)+7q&=0\\ 10r+p&=5q \end{aligned} Yes, because all three equations have three variables, $$p$$, $$q$$, and $$r$$, of power $$1$$.

 Relationship of Systems of Linear Equations in Three Variables with Axis A system of linear equations in three variables has three axes, namely the $$x$$-axis, $$y$$-axis and $$z$$-axis. All three linear equations form a plane on each axis. Every linear equation in two variables forms a straight line on each axis.

 Form in Cartesian Plane Geometrically, a linear equation in three variables forms a plane in a three-dimensional space.

 Type of Solution Description One Solution The planes intersect at only one point Infinite Solutions The planes intersect in a straight line No Solution The planes do not intersect at any point

 Methods used to solve systems of linear equations in three variables Elimination Method Substitution Method

Example $$1$$
 Question

Solve the following system of linear equations using the elimination method.

\begin{aligned} 4x-3y+z&=-10 \\ 2x+y+3z&=0\\ -x+2y-5z&=17 \end{aligned}

 Solution

Choose any two equations.

\begin{aligned} 4x-3y+z&=-10 \quad \cdots\boxed{1} \\ 2x+y+3z&=0 \quad \quad \,\, \cdots \boxed{2} \end{aligned}

Multiply equation $$\boxed{2}$$ with $$2$$ so that the coefficients of $$x$$ are equal.

$$\boxed{2}\times 2:\quad 4x+2y+6z=0 \quad \cdots \boxed{3}.$$

Eliminate the variable $$x$$ by subtracting $$\boxed{1}$$ from $$\boxed{3}$$.

$$\boxed{3}-\boxed{1}:\quad 5y+5z=10 \quad \cdots \boxed{4}.$$

Choose another set of two equations.

\begin{aligned} 2x+y+3z&=0 \quad\,\,\, \cdots \boxed{5} \\ -x+2y-5z&=17 \quad \cdots\boxed{6} \end{aligned}

Multiply equation $$\boxed{6}$$ with $$2$$ so that the coefficients of variable $$x$$ are equal.

\begin{aligned} \boxed{6}\times2:-2x+4y-10z&=34 \quad\cdots\boxed{7}\\ \boxed{5}+\boxed{7}: \quad\quad\quad 5y-7z&=34 \quad\cdots\boxed{8} \\ \boxed{4}-\boxed{8}: \quad\quad\quad\quad\,\,\,\ 12z&=-24 \\ z&=-2. \end{aligned}

Substitute $$z=-2$$ into $$\boxed{8}$$.

\begin{aligned} 5y-7(-2)&=34 \\ 5y+14&=34\\ 5y&=20 \\ y&=4. \end{aligned}

Substitute $$y=4$$ and $$z=-2$$ into $$\boxed{1}$$.

\begin{aligned} 4x-3(4)+(-2)&=-10\\ 4x-12-2&=-10\\ 4x&=4\\ x&=1. \end{aligned}

Thus, $$x=1$$$$y=4$$, and $$z=-2$$ are the solutions to this system of linear equations.

Example $$2$$
 Question

Solve the following system of linear equations using the substitution method.

\begin{aligned} 3x-y-z&=-120 \\ y-2z&=30 \\ x+y+z&=180 \end{aligned}

 Solution

Label all the equations.

\begin{aligned} 3x-y-z&=-120 \quad \cdots \boxed{1} \\ y-2z&=30 \quad\quad\,\, \cdots \boxed{2} \\ x+y+z&=180 \quad\,\,\,\ \cdots \boxed{3} \end{aligned}

From $$\boxed{1}$$$$z=3x-y+120 \quad\cdots\boxed{4}$$.

Substitute $$\boxed{4}$$ into $$\boxed{2}$$.

\begin{aligned} y-2(3x-y+120)&=30 \\ y-6x+2y-240&=30 \\ -6x+3y&=270 \\ y&=90+2x \quad \cdots \boxed{5}. \end{aligned}

Substitute $$\boxed{4}$$ and $$\boxed{5}$$ into $$\boxed{3}$$.

\begin{aligned} x+(90+2x)+[3x-(90+2x)+120]&=180 \\ x+2x+3x-2x+90-90+120&=180 \\ 4x&=60 \\ x&=15. \end{aligned}

Substitute $$x=15$$ into $$\boxed{5}$$.

\begin{aligned} y&=90+2(15) \\ &=120. \end{aligned}

Substitute $$x=15$$ and $$y=120$$ into $$\boxed{3}$$.

\begin{aligned} 15+120+z&=180 \\ z&=45. \end{aligned}

Thus, $$x=15$$$$y=120$$, and $$z=45$$ are the solutions to this system of linear equations.

## Systems of Linear Equations in Three Variables

 3.1 Systems of Linear Equations in Three Variables

 Definition Two or more linear equations involving the same set of variables form a system of linear equations.

 General Form of a Linear Equation in Three Variables The general form of a linear equation in three variables can be written as follows: $$ax+by+cz=d$$, where $$a$$, $$b$$, and $$c$$ are not equal to zero.

Example of Systems of Linear Equations in Three Variables
 System of Equations Description \begin{aligned} 2x+4y-z&=10\\ x+y&=10z^2\\ 5y-z-2x&=3 \end{aligned} Not systems of linear equations, because there is an equation in which the highest power of the variable is $$2$$. \begin{aligned} p+8q-4r&=2\\ 2(p+6r)+7q&=0\\ 10r+p&=5q \end{aligned} Yes, because all three equations have three variables, $$p$$, $$q$$, and $$r$$, of power $$1$$.

 Relationship of Systems of Linear Equations in Three Variables with Axis A system of linear equations in three variables has three axes, namely the $$x$$-axis, $$y$$-axis and $$z$$-axis. All three linear equations form a plane on each axis. Every linear equation in two variables forms a straight line on each axis.

 Form in Cartesian Plane Geometrically, a linear equation in three variables forms a plane in a three-dimensional space.

 Type of Solution Description One Solution The planes intersect at only one point Infinite Solutions The planes intersect in a straight line No Solution The planes do not intersect at any point

 Methods used to solve systems of linear equations in three variables Elimination Method Substitution Method

Example $$1$$
 Question

Solve the following system of linear equations using the elimination method.

\begin{aligned} 4x-3y+z&=-10 \\ 2x+y+3z&=0\\ -x+2y-5z&=17 \end{aligned}

 Solution

Choose any two equations.

\begin{aligned} 4x-3y+z&=-10 \quad \cdots\boxed{1} \\ 2x+y+3z&=0 \quad \quad \,\, \cdots \boxed{2} \end{aligned}

Multiply equation $$\boxed{2}$$ with $$2$$ so that the coefficients of $$x$$ are equal.

$$\boxed{2}\times 2:\quad 4x+2y+6z=0 \quad \cdots \boxed{3}.$$

Eliminate the variable $$x$$ by subtracting $$\boxed{1}$$ from $$\boxed{3}$$.

$$\boxed{3}-\boxed{1}:\quad 5y+5z=10 \quad \cdots \boxed{4}.$$

Choose another set of two equations.

\begin{aligned} 2x+y+3z&=0 \quad\,\,\, \cdots \boxed{5} \\ -x+2y-5z&=17 \quad \cdots\boxed{6} \end{aligned}

Multiply equation $$\boxed{6}$$ with $$2$$ so that the coefficients of variable $$x$$ are equal.

\begin{aligned} \boxed{6}\times2:-2x+4y-10z&=34 \quad\cdots\boxed{7}\\ \boxed{5}+\boxed{7}: \quad\quad\quad 5y-7z&=34 \quad\cdots\boxed{8} \\ \boxed{4}-\boxed{8}: \quad\quad\quad\quad\,\,\,\ 12z&=-24 \\ z&=-2. \end{aligned}

Substitute $$z=-2$$ into $$\boxed{8}$$.

\begin{aligned} 5y-7(-2)&=34 \\ 5y+14&=34\\ 5y&=20 \\ y&=4. \end{aligned}

Substitute $$y=4$$ and $$z=-2$$ into $$\boxed{1}$$.

\begin{aligned} 4x-3(4)+(-2)&=-10\\ 4x-12-2&=-10\\ 4x&=4\\ x&=1. \end{aligned}

Thus, $$x=1$$$$y=4$$, and $$z=-2$$ are the solutions to this system of linear equations.

Example $$2$$
 Question

Solve the following system of linear equations using the substitution method.

\begin{aligned} 3x-y-z&=-120 \\ y-2z&=30 \\ x+y+z&=180 \end{aligned}

 Solution

Label all the equations.

\begin{aligned} 3x-y-z&=-120 \quad \cdots \boxed{1} \\ y-2z&=30 \quad\quad\,\, \cdots \boxed{2} \\ x+y+z&=180 \quad\,\,\,\ \cdots \boxed{3} \end{aligned}

From $$\boxed{1}$$$$z=3x-y+120 \quad\cdots\boxed{4}$$.

Substitute $$\boxed{4}$$ into $$\boxed{2}$$.

\begin{aligned} y-2(3x-y+120)&=30 \\ y-6x+2y-240&=30 \\ -6x+3y&=270 \\ y&=90+2x \quad \cdots \boxed{5}. \end{aligned}

Substitute $$\boxed{4}$$ and $$\boxed{5}$$ into $$\boxed{3}$$.

\begin{aligned} x+(90+2x)+[3x-(90+2x)+120]&=180 \\ x+2x+3x-2x+90-90+120&=180 \\ 4x&=60 \\ x&=15. \end{aligned}

Substitute $$x=15$$ into $$\boxed{5}$$.

\begin{aligned} y&=90+2(15) \\ &=120. \end{aligned}

Substitute $$x=15$$ and $$y=120$$ into $$\boxed{3}$$.

\begin{aligned} 15+120+z&=180 \\ z&=45. \end{aligned}

Thus, $$x=15$$$$y=120$$, and $$z=45$$ are the solutions to this system of linear equations.