Solve

\[\begin{aligned} x+y &= 4\\ x-y &= -2\end{aligned}\]

Solve
\(x+y=2\)
\(y=2x-1\)

Solve
\(x-3y=-1\)
\(2x-3y=4\)

Solve
\(4x+2y=8\)
\(y=-2x+4\)

Solve by substitution.

\[\begin{aligned} 2x+6y&=7\\ x&=-3y+5\end{aligned}\]

Solve

\[\begin{aligned} x+y&=4\\ x-y&=2\end{aligned}\]

Solve

\[\begin{aligned} 2x-y&=6\\ x+3y&=3\end{aligned}\]

Solve

\[\begin{aligned} \displaystyle\frac{1}{2}x-\displaystyle\frac{1}{3}y&=2\\ \displaystyle\frac{1}{4}x+\displaystyle\frac{2}{3}y&=6\end{aligned}\]

Solve

\[\begin{aligned} 2x-y&=2\\ 4x-2y&=12\end{aligned}\]

Solve

\[\begin{aligned} 4x-3y&=2\\ 8x-6y&=4\end{aligned}\]

Solve
\(\begin{align*} x+y+z &= 6\\ 2x-y+z &= 3\\ x+2y-3z &= -4 \end{align*}\)

Solve
\(\begin{align} 2x+y-z&=3\\ 3x+4y+z&=6\\ 2x-3y+z&=1 \end{align}\)

Solve
\(\begin{align} 2x+3y-z&=5\\ 4x+6y-2&=10\\ x-4y+3z&=5 \end{align}\)

Solve
\(\begin{align}x-5y+4z&=8\\ 3x+y-2z&=7\\ -9x-3y+6z&=5 \end{align}\)

Solve
\(\begin{align} x+3y&=5\\ 6y+z&=12\\ x-2z&=-10 \end{align}\)

Solve using an augmented matrix.

\[\begin{aligned} x+y-z &= 2\\ 2x+3y-z &= 7\\ 3x-2y+z &= 9\end{aligned}\]

Find the value of each:

1. \(\left|\begin{array}{cc} 1 & 2\\ 3 & 4 \end{array}\right|\)

2. \(\left|\begin{array}{cc} 3 & -2\\ 5 & 7 \end{array}\right|\)

Solve for \(x\): \(\begin{vmatrix} x&2\\ x&4 \end{vmatrix}=8\)

Use Cramer’s rule to solve

\[\begin{aligned} 2x-3y &= 4\\ 4x+5y &=3\end{aligned}\]

Find the value of \(\begin{vmatrix} 1&3&-2\\ 2&0&1\\ 4&-1&1 \end{vmatrix}\)

Expand across the first row: \(\begin{vmatrix}1&3&-2\\ 2&0&1\\ 4&-1&1\end{vmatrix}\)

Expand down column two: \(\begin{vmatrix}2&3&-2\\ 1&4&1\\ 1&5&-1\end{vmatrix}\)

Use Cramer’s Rule to solve

\[\begin{aligned} x+y+z &= 6\\ 2x-y+z &= 3\\ x+2y-3z &=-4\end{aligned}\]

Use Cramer’s Rule to solve

\[\begin{aligned} x+y &= -1\\ 2x-z &= 3\\ y+2z &=-1\end{aligned}\]

One number is \(2\) more than \(3\) times another. Their sum is \(26\). Find the two numbers.

Suppose \(850\) tickets were sold for a game for a total of \(\$1\text{,}100\). If adult tickets cost \(\$1.50\) and children’s tickets cost \(\$1.00\), how many of each ticket were sold?

A person invests \(\$10\text{,}000\) in two accounts. One account earns \(8\%\) annually and the other earns \(9\%\). If the total interest earned from both accounts in a year is \(\$860\), how much was invested in each account?

How much \(20\%\) alcohol and \(50\%\) alcohol must be mixed to get \(12\) gallons of \(30\%\) alcohol solution?

It takes \(2\) hours for a boat to travel \(28\) miles downstream. The same boat can travel \(18\) miles upstream in \(3\) hours. What is the speed of the boat in still water, and what is the speed of the current of the river?

A coin collection consists of \(14\) coins with a total value of \(\$1.35\). If the coins are nickels, dimes, and quarters, and the number of nickels is \(3\) less than twice the number of dimes, how many of each coin is there?

Graph
\(\begin{align}y &< \displaystyle\frac{1}{2}x+3\\ y &\geq \displaystyle\frac{1}{2}x-2 \end{align}\)

Graph
\(\begin{align}x+y &<4\\ x &\geq 0\\ y &\geq 0 \end{align}\)

Graph
\(\begin{align}x&\leq 4\\ y&\geq -3 \end{align}\)

Graph

\[\begin{aligned} x-2y&\leq 4\\ x+y&\leq 4\\ x&\geq -1\end{aligned}\]

A basketball arena charges \(\$20\) for certain seats and \(\$15\) for others. They want to make more than \(\$18,000\) and reserve at least \(500\) \(\$15\) seats. Find the system of inequalities and sketch the graph. If \(620\) tickets are sold for \(\$15\), at least how many are sold for \(\$20\)?

Make-A-Sale is a new company designed to assist clients selling household items over the internet. Start-up costs for the company are \(\$40,000\), and they expect variable costs to be \(\$6.00\) for each item sold. They plan to charge \(\$14.00\) per item.

1. Find the total cost, revenue, and profit for selling \(x\) items.

2. If they sell \(3,000\) items, find their cost, revenue, and profit.

3. If they sell \(12,000\) items, find their cost, revenue, and profit.

4. Find the break-even point (when revenue equals cost), include a graph of the inequalities \(R(x)\leq C(x)\) and \(R(x)\geq C(x)\) to show intervals of profit and loss.

Find the equilibrium price and quantity of the following supply and demand equations: \[d(p)=5,000-4p\] \[s(p)=1,000+6p\]