Systems of Linear Equations

Write the system of equations that corresponds to the augmented matrix. 

$\left[\begin{array}{ccc}2& -4& -2\\ 3& -3& -1\end{array}\right]$

Write the system of equations that corresponds to the augmented matrix.  

$\left[\begin{array}{rrrr}1& 0& -3& -1\\ 1& -2& 0& -2\\ 0& -1& 2& 3\end{array}\right]$

In the following exercise, perform the indicated operations on the augmented matrices.

$\left[\begin{array}{rrrr}1& -3& -2& 4\\ 2& 2& -1& -3\\ 4& -2& -3& -1\end{array}\right]$

ⓐ Interchange rows $2$ and $3$.

ⓑ Multiply row $1$ by $2$.

ⓒ Multiply row $3$ by $-2$ and add

to row $2$.

In the following exercises, solve each system of equations using a matrix.

$\left\{\begin{array}{l}4x + y = 6\\ x - y = 4\end{array}\right.$

In the following exercise, perform the indicated operations on the augmented matrices.

$\left[\begin{array}{lll}4& -6& -3\\ 3& 2& 1\end{array}\right]$

ⓐ Interchange rows $2$ and $1$.

ⓑ Multiply row $1$ by $4$.

ⓒ Multiply row $2$ by $3$ and add to row $1$.

In the following exercise, solve each system of equations using a matrix.

$\left\{\begin{array}{l}2y+3z=-1\\ 5x+3y=-6\\ 7x+z=1\end{array}\right.$ 

In the following exercise, solve each system of equations using a matrix. 

$\left\{\begin{array}{l}x+2y-3z=-1\\ x-3y+z=1\\ 2x-y-2z=2\end{array}\right.$

In the following exercise, solve each system of equations using a matrix.

$\left\{\begin{array}{l}x+y-3z=-1\\ y-z=0\\ -x+2y=1\end{array}\right.$

In the following exercise, evaluate the determinate of the square matrix.

$\left[\begin{array}{cc}8& -4\\ 5& -3\end{array}\right]$

In the following exercise, find and then evaluate the indicated minors

$\left|\begin{array}{rrr}-1& -3& 2\\ 4& -2& -1\\ -2& 0& -3\end{array}\right|$

Find the minor

a. $a_1$ b. $b_1$ c. $c_2$

In the following exercise, evaluate each determinant by expanding by minors along the first row.

$\left|\begin{array}{rrr}-2& -3& -4\\ 5& -6& 7\\ -1& 2& 0\end{array}\right|$

In the following exercise, evaluate each determinant by expanding by minors.

$\left|\begin{array}{rrr}3& 5& 4\\ -1& 3& 0\\ -2& 6& 1\end{array}\right|$

In the following exercises, solve each system of equations using Cramer’s rule

$\left\{\begin{array}{l}x-3y=-9\\ 2x+5y=4\end{array}\right.$

In the following exercise, solve each system of equations using Cramer’s rule

$\left\{\begin{array}{l}4x-3y+z=7\\ 2x-5y-4z=3\\ 3x-2y-2z=-7\end{array}\right.$

In the following exercise, solve each system of equations using Cramer’s rule

$\left\{\begin{array}{l}2x+5y=4\\ 3y-z=3\\ 4x+3z=-3\end{array}\right.$

In the following exercise, solve the system of equations using Cramer’s rule

$\left\{\begin{array}{l}x+y-3z=-1\\ y-z=0\\ -x+2y=1\end{array}\right.$

In the following exercise, solve the system of equations using Cramer’s rule

$\left\{\begin{array}{l}3x+4y-3z=-2\\ 2x+3y-z=-12\\ x+y-2z=6\end{array}\right.$

In the following exercises, determine whether the given points are collinear.

$(0,2),(-1,-1),(-2,4)$

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities
In the following exercises, determine whether each ordered pair is a solution to the system. 

$\left\{\begin{array}{l}4x+y>6\\ 3x-y\le 12\end{array}\right.$

(a) $(2,-1)$

(b) $(3,-2)$

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities
In the following exercises, determine whether each ordered pair is a solution to the system.

$\left\{\begin{array}{l}y>\frac{1}{3}x+2\\ x-\frac{1}{4}y\le 10\end{array}\right.$

(a) $(6,5)$

(b) $(15,8)$

Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.

$\left\{\begin{array}{l}y<3x+1\\ y\ge -x-2\end{array}\right.$

Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.

$\left\{\begin{array}{l}x-y>-1\\ y<\frac{1}{3}x-2\end{array}\right.$

Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.

$\left\{\begin{array}{l}2x-3y<6\\ 3x+4y\ge 12\end{array}\right.$

Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.

$\left\{\begin{array}{l}y\le -\frac{3}{4}x+1\\ x\ge -5\end{array}\right.$

Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.

$\left\{\begin{array}{l}x+3y<5\\ y\ge -\frac{1}{3}x+6\end{array}\right.$

Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.

$\left\{\begin{array}{l}y\ge 2x-5\\ -6x+3y>-4\end{array}\right.$

Solve Applications of Systems of Inequalities
In the following exercises, translate to a system of inequalities and solve.

Roxana makes bracelets and necklaces and sells them at the farmers’ market. She sells the bracelets for $\backslash (12$ each and the necklaces for $\backslash )18$ each. At the market next weekend she will have room to display no more than $40$ pieces, and she needs to sell at least $\$500$ worth in order to earn a profit.

(a) Write a system of inequalities to model this
situation.
(b) Graph the system.
(c) Should she display $26$ bracelets and $14$ necklaces?
(d) Should she display $39$ bracelets and $1$ necklace?

Solve Applications of Systems of Inequalities
In the following exercises, translate to a system of inequalities and solve.

Annie has a budget of $\backslash (600$ to purchase
paperback books and hardcover books for her
classroom. She wants the number of hardcover to be at least $5$ more than three times the number of paperback books. Paperback books cost $\backslash )4$ each and hardcover
books cost $\$15$ each.
(a) Write a system of inequalities to model this
situation.
(b) Graph the system.
(c) Can she buy $8$ paperback books and $40$ hardcover books?
(d) Can she buy $10$ paperback books and $37$ hardcover books?

In the following exercises, solve the following systems by graphing.

$\left\{\begin{array}{l}x-y=5\\ x+2y=-4\end{array}\right.$

In the following exercises, solve the following systems by graphing.

$\left\{\begin{array}{l}x-y>-2\\ y\le 3x+1\end{array}\right.$

In the following exercises, solve each system of equations. Use either substitution or elimination.

$\left\{\begin{array}{l}x+4y=6\\ -2x+y=-3\end{array}\right.$

In the following exercises, solve each system of equations. Use either substitution or elimination.

$\left\{\begin{array}{l}-3x+4y=25\\ x-5y=-23\end{array}\right.$

In the following exercises, solve each system of equations. Use either substitution or elimination.

$\left\{\begin{array}{l}x+y-z=-1\\ 2x-y+2z=8\\ -3x+2y+z=-9\end{array}\right.$

Solve the system of equations using a matrix.

$\left\{\begin{array}{l}2x+y=7\\ x-2y=6\end{array}\right.$

Solve the system of equations using a matrix.

$\left\{\begin{array}{l}-3x+y+z=-4\\ -x+2y-2z=1\\ 2x-y-z=-1\end{array}\right.$

Solve using Cramer’s rule.

$\left\{\begin{array}{l}3x+y=-3\\ 2x+3y=6\end{array}\right.$

Evaluate the determinant by

expanding by minors:

$\left|\begin{array}{ccc}3& -2& -2\\ 2& -1& 4\\ -1& 0& -3\end{array}\right|$

Greg is paddling his canoe

upstream, against the current, to

a fishing spot 10 miles away. If he

paddles upstream for 2.5 hours

and his return trip takes 1.25

hours, find the speed of the current

and his paddling speed in still

water.

A pharmacist needs 20 liters

of a 2% saline solution. He has a 1%

and a 5% solution available. How

many liters of the 1% and how

many liters of the 5% solutions

should she mix to make the 2%

solution?

Arnold invested \(64,000,

some at 5.5% interest and the rest

at 9%. How much did he invest at

each rate if he received \)4,500 in

interest in one year?

The church youth group is

selling snacks to raise money to

attend their convention. Amy sold

2 pounds of candy, 3 boxes of

cookies and 1 can of popcorn for

a total sales of \(65. Brian sold 4

pounds of candy, 6 boxes of

cookies and 3 cans of popcorn for

a total sales of \)140. Paulina sold

8 pounds of candy, 8 boxes of

cookies and 5 can of popcorn for a

total sales of $250. What is the cost

of each item?

The manufacturer of a

granola bar spends \(1.20 to make

each bar and sells them for \)2. The

manufacturer also has fixed costs

each month of $8,000.(

a) Find the cost function C when x

granola bars are manufactured

(b) Find the revenue function R

when x granola bars are sold.

(c) Show the break-even point by

graphing both the Revenue and

Cost functions on the same grid.

(d) Find the break-even point.

Interpret what the break-even

point means.

Translate to a system of

inequalities and solve.

Andi wants to spend no more than

\(50 on Halloween treats. She

wants to buy candy bars that cost

\)1 each and lollipops that cost

$0.50 each, and she wants the

number of lollipops to be at least

three times the number of candy

bars.

(a) Write a system of inequalities to

model this situation.

(b) Graph the system.

(c) Can she buy 20 candy bars and

40 lollipops?