4.07 Solving linear systems in three unknowns
Determine whether each ordered triple is a solution of the corresponding system of equations:
Triple: \left( - 9 , - 7 , 4\right)
System:
\begin{aligned}-2x+5y-3z&=-29\\-3x+y+4z&=38\\-4x+2y+3z&=34\end{aligned}
Triple: \left(1, - 5 , - 3 \right)
System:
\begin{aligned}4x-5y+3z&=20\\4x-5y-z&=32\\-x+2y-5z&=4\end{aligned}
Triple: \left(\dfrac{3}{4}, \dfrac{4}{5}, \dfrac{2}{5}\right)
System:
\begin{aligned}5x+y+3z&=\dfrac{23}{4}\\4x+3y+5z&=\dfrac{37}{5}\\2x-4y+3z&=-\dfrac{1}{2}\end{aligned}
Triple: \left(1.1, 0.3, 1.7\right)
System:
\begin{aligned}3x+y-5z&=-5.9\\5x+y-4z&=-1\\5x-2y-4z&=-1.9\end{aligned}
Triple: \left( - 8 , - 1 , 2\right)
System:
\begin{aligned}x+5y+4z&=-8\\4x-y+3z&=-25\\2y-z&=-4\end{aligned}
Triple: \left( - 6 , 3, - 9 \right)
System:
\begin{aligned}x+5y+3z&=-18\\5x+y-3z&=-3\\5x+3y-z&=-12\end{aligned}
Solve the following systems of equations:
\begin{aligned}x+y&=6 \\ y+z&=13 \\ x+z&=9 \end{aligned}
\begin{aligned} 3x+2y-z &= -9 \\ 6x-4y+z &= -20 \\ 3x+6y+4z&=5 \end{aligned}
How many solutions do each of the following systems of equations have?
\begin{aligned}3x+4y+2z&=-1\\5x-2y-3z&=-23\\-3x-4y-2z&=1\end{aligned}
\begin{aligned}6x+7y-4z&=-13\\12x+14y-8z&=-24\\3x-2y+5z&=17\end{aligned}
Consider the system:
| \text{Equation 1} | \text{Equation 2} |
| 2 x - 3 y - z = - 19 | y - z = - 1 |
Let z = t be arbitrary.
Solve Equation 2 for y with respect to t.
Using your result from part a, solve Equation 1 for x with respect to t.
Hence, write down the solution to the system with respect to the arbitrary variable t.
Consider the system:
| \text{Equation 1} | \text{Equation 2} |
| 4 x +5y - z = 11 | x+y +2 z = 9 |
Let z = t be arbitrary.
Solve Equation 2 for y with respect to x and t.
Solve Equation 1 for x with respect to t.
Hence, solve for y with respect to t.
Hence, write down the solution to the system with respect to the arbitrary variable t.
Consider the system:
| \text{Equation 1} | \text{Equation 2} |
| x-3y +3 z = 6 | x-2y +4z = 5 |
Let z = t be arbitrary.
Solve Equation 1 for x with respect to y and t.
Solve Equation 2 for y with respect to t.
Hence, solve for x with respect to t.
Hence, write down the solution to the system with respect to the arbitrary variable t.
Consider the following system of three equations:
\begin{aligned}3x+5y+8z&=136\\27x+45y+72z&=872\\x+y+z&=25\end{aligned}
Does the system have one solution, infinite solution or no solution?
Is the system consistent or incosistent?
Consider the following system of 3 equations:
\begin{aligned}8x+5y+8z&=97\\6x+y+4z&=45\\x+y+z&=14\end{aligned}
Solve the system.
Is the system independent or dependent?
Consider the following system of 3 equations:
\begin{aligned}7x+3y+3z&=3.2\\8x+5y+3z&=4.4\\4x+9y+4z&=5.7\end{aligned}
Solve the system.
Is the system consistent or incosistent?
Find a system of three linear equations in three variables that has \left( - 4 , - 2 , -1\right) as a solution.
Solve the following systems of equations:
For each of the following system of 3 equations:
Solve the system.
Determine whether the system is consistent or inconsistent.
Determine whether the system is dependent or independent.
Consider the equation: A x + B y + C z = 12.
One solution to this equation is \left(1, \dfrac{3}{4}, 3\right). Substitute these values into the equation, to give an equation in terms of A, B, and C.
A second solution to the equation is \left(\dfrac{4}{3}, 1, 2\right). Substitute these values into the equation, to give an equation in terms of A, B, and C.
A third solution to the equation is \left(2, 1, 1\right). Substitute these values into the equation, to give an equation in terms of A, B, and C.
Solve your three equations for A, B, and C.
The fraction \dfrac{1}{24} can be written as the following sum:
\dfrac{1}{24} = \dfrac{x}{8} + \dfrac{y}{4} + \dfrac{z}{3}where the numbers x, y, and z are solutions of:
\begin{aligned}x +y+ z&=1------\text{ equation 1}\\2x-y+z&=0------\text{ equation 2}\\-x+2y+2z&=-1-----\text{ equation 3}\end{aligned}Solve this system of equations for x, \, y, and z.
Use your values of x, \, y, and z to determine whether the following equation is true or false:
\dfrac{1}{24} = \dfrac{x}{8} + \dfrac{y}{4} + \dfrac{z}{3}Consider the quadratic with equation y = a x^{2} + b x + c that passes through the points \left(1, 15\right), \, \left(2, 5\right) and \left(4, - 3 \right).
Set up a system of equations in the variables a, b and c using the points given.
Solve your equations for a, \, b, and c.
Write down the quadratic equation that passes through the points \left(1, 15\right), \left(2, 5\right) and \left(4, - 3 \right).
The equation of a circle can be written in the general form x^{2} + y^{2} + a x + b y + c = 0. The circle below passes through the points \left(2, - 4 \right), \left(5, 1\right) and \left(3, - 3 \right):
The general form of a circle given may be rearranged to the following form: \\ a x + b y + c = - \left( x^{2} + y^{2} \right). Use this form to set up a system of equations in the variables a, b and c from the given points.
Solve your equations for a, \, b, and c.
Write down the equation of the circle that passes through the points \left(2, - 4 \right), \left(5, 1\right) and \left(3, - 3 \right).