Exam 43: Multivariable Linear Systems

Find values of x,y,and ? that satisfy the system.These systems arise in certain optimization problems in calculus,and ? is called a Lagrange multiplier.? $\left\{\begin{array}{ll}y+\lambda & =0 \\x+\lambda & =0 \\x+y-18 & =0\end{array}\right.$ ?

A residential building contractor borrowed $29,000 to complete a new home.Some of the money was borrowed at 4%,some at 6%,and some at 7%.How much was borrowed at each rate if the annual interest owed was $1,580 and the amount borrowed at 6% is two times more than the amount borrowed at 7%?

Solve the system of linear equations and check any solution algebraically.? $\left\{ \begin{array} { l } 5 x + 3 y + 17 z = 0 \\6 x + 4 y + 22 z = 0 \\5 x + 2 y + 19 z = 0\end{array} \right.$ ?

Determine whether the ordered triple is a solution of the system of equations. ? ?? $\left\{ \begin{array} { c l } 6 x - y + 2 z & = 26 \\4 x - 3 z & = 26 \\2 x + 5 z & = 0\end{array} \right.$ ?? $( 5,0 , - 2 )$ ?

Write the partial fraction decomposition of the rational expression.?? $\frac { 3 } { x ^ { 2 } + 5 x + 4 }$

A small corporation borrowed $800,000 to expand its line of toys.Some of the money was borrowed at 8%,some at 9%,and some at 10%.How much was borrowed at each rate if the annual interest owed was $67,400 and the amount borrowed at 8% was five times the amount borrowed at 10%? ​

Find the value of x,y and z in the figure.​ ​ where a = 6,b = 14 ​

Solve the system,if possible. ?? $\left\{ \begin{array} { l l } x + y + z & = - 20 \\2 x + y + z & = - 30 \\5 x + y - z & = - 46\end{array} \right.$ ?

An object moving vertically is at the given heights at the specified times.Find the position equation $s = \frac { 1 } { 2 } a t ^ { 2 } + v _ { 0 } t + s _ { 0 }$ for the object. At t = 1 second,s = 299 feet At t = 2 seconds,s = 281 feet At t = 3 seconds,s = 231 feet

Find the equation of the parabola $y = a x ^ { 2 } + b x + c$ that passes through the points. ?? $( 0,5 ) , ( 1,6 ) , ( 2,5 )$ ?

A mixture of 7 pounds of fertilizer A,21 pounds of fertilizer B,and 8 pounds of fertilizer C provides the optimal nutrients for a plant.Commercial brand X contains equal parts of fertilizer B and fertilizer C.Commercial brand Y contains one part of fertilizer A and two parts of fertilizer B.Commercial brand Z contains two parts of fertilizer A,five parts of fertilizer B,and two parts of fertilizer C.How much of each fertilizer brand is needed to obtain the desired mixture? ​

?Applying Kirchhoff's Laws to the electrical network in the figure,the currents I₁,I₂,and I₃ are the solution of the system $\left\{ \begin{array} { l l } I _ { 1 } - I _ { 2 } + I _ { 3 } & = 0 \\6 I _ { 1 } + 3 I _ { 2 } & = 11 \\3 I _ { 2 } + 5 I _ { 3 } & = 6\end{array} \right.$ Find the currents.? =11 volts, =6 volts =6\Omega,=3\Omega,=5\Omega ?

Use back-substitution to solve the system of linear equations.? $\left\{ \begin{array} { c c } 2 x + y - 3 z & = 10 \\y + z & = 6 \\z & = 4\end{array} \right.$ ?

You work as a disc jockey at your college radio station.You are supposed to play 44 songs within two hours.You are to choose the songs from the latest rock,dance,and pop albums.You want to play twice as many rock songs as pop songs and four more pop songs than dance songs.How many of each type of song will you play? ​