In Exercises reviewsystemfirst - reviewsystemlast, solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically.
\(\left \{ \begin{array}{rcr} x+2y & = & 5 \\ x & = & 6 \end{array} \right .\)

The system is:

consistent independent consistent dependent inconsistent

\(\left (6, -\frac {1}{2}\right )\)
\(\left \{ \begin{array}{rcr} 2y-3x & = & 1 \\ y & = & -3 \end{array} \right .\)

The system is:

consistent independent consistent dependent inconsistent

\(\left (-\frac {7}{3}, -3\right )\)
\(\left \{ \begin{array}{rcr} \frac {x+2y}{4} & = & -5 \\ \frac {3x-y}{2} & = & 1 \end{array} \right .\)

The system is:

consistent independent consistent dependent inconsistent

\(\left (-\frac {16}{7}, -\frac {62}{7}\right )\)
\(\left \{\begin{array}{ccccc} \frac {2}{3} x & - &\frac {1}{5}y&= &3 \\ \frac {1}{2}x& +&\frac {3}{4}y&=&1 \end{array}\right .\)

The system is:

consistent independent consistent dependent inconsistent

\(\left (\frac {49}{12}, -\frac {25}{18}\right )\)
\(\left \{ \begin{array}{ccccr} \frac {1}{2}x & -& \frac {1}{3}y & = & -1 \\ 2y&-&3x & = & 6 \end{array} \right .\)

The system is:

consistent independent consistent dependent inconsistent

\(\left (t, \frac {3}{2}t+3\right )\) for all real numbers \(t\)
\(\left \{ \begin{array}{ccccr} x&+&4y & = & 6 \\ \frac {1}{12}x&+&\frac {1}{3}y& = & \frac {1}{2} \end{array} \right .\)

The system is:

consistent independent consistent dependent inconsistent

\(\left (6-4t, t\right )\) for all real numbers \(t\)
\(\left \{ \begin{array}{ccccr} 3y&-&\frac {3}{2}x & = & -\frac {15}{2} \\ \frac {1}{2}x&-&y & = & \frac {3}{2} \end{array} \right .\)

The system is:

consistent independent consistent dependent inconsistent

No solution
\(\left \{ \begin{array}{ccccr} \frac {5}{6}x&+&\frac {5}{3}y & = & -\frac {7}{3} \\ -\frac {10}{3}x&-&\frac {20}{3}y & = & 10 \end{array} \right .\)

The system is:

consistent independent consistent dependent inconsistent

No solution
A local buffet charges \(\$7.50\) per person for the basic buffet and \(\$9.25\) for the deluxe buffet (which includes crab legs.) If 27 diners went out to eat and the total bill was \(\$227.00\) before taxes, how many chose the basic buffet and how many chose the deluxe buffet?

\(13\) chose the basic buffet and \(14\) chose the deluxe buffet.
At The Old Home Fill’er Up and Keep on a-Truckin’ Cafe, Mavis mixes two different types of coffee beans to produce a house blend. The first type costs $3 per pound and the second costs $8 per pound. How much of each type does Mavis use to make 50 pounds of a blend which costs $6 per pound?

Mavis needs 20 pounds of $3 per pound coffee and 30 pounds of $8 per pound coffee.
Skippy has a total of \(\$\)10,000 to split between two investments. One account offers \(3\%\) simple interest, and the other account offers \(8\%\) simple interest. For tax reasons, he can only earn \(\$500\) in interest the entire year. How much money should Skippy invest in each account to earn \(\$500\) in interest for the year?

Skippy needs to invest \(\$\)6000 in the \(3\%\) account and \(\$\)4000 in the \(8 \%\) account.
A \(10 \%\) salt solution is to be mixed with pure water to produce 75 gallons of a \(3\%\) salt solution. How much of each are needed?

\(22.5\) gallons of the \(10 \%\) solution and \(52.5\) gallons of pure water.
This exercise is a follow-up to Example dudebromixture. Work with your classmates to explain why mixing 4 gallons of Sasquatch SweatTM Energy Drink and 1 gallon of Frooty Giggle DelightTM would also produce a mixture that was “close enough for the Dude-Bros”.