11.7: Exercise Supplement

Exercise Supplement

Solutions by Graphing - Elimination by Addition

For the following problems, solve the systems of equations.

Exercise $\PageIndex{1}$

$\left\{\begin{array}{r} 4x + y = 5\\ -2x + 3y = -13 \end{array}\right.$

Answer

$(2, -3)$

Exercise $\PageIndex{2}$

$\left\{\begin{array}{r} -5x + 2y = 5\\ x + 7y = -1 \end{array}\right.$

Exercise $\PageIndex{3}$

$\left\{\begin{array}{r} x - 3y = 17\\ 8x + 2y = 46 \end{array}\right.$

Answer

$(\dfrac{86}{13}, -\dfrac{45}{13})$

Exercise $\PageIndex{4}$

$\left\{\begin{array}{r} 6m + 5n = -9\\ 2m - 4n = 14 \end{array}\right.$

Exercise $\PageIndex{5}$

$\left\{\begin{array}{r} 3x - 9y = 5\\ -x + 3y = 0 \end{array}\right.$

Answer

no solution

Exercise $\PageIndex{6}$

$\left\{\begin{array}{r} y = 2x - 5\\ 8x - 75 = 5 \end{array}\right.$

Exercise $\PageIndex{7}$

$\left\{\begin{array}{r} x = 8\\ 9y = 5x - 76 \end{array}\right.$

Answer

$(8, -4)$

Exercise $\PageIndex{8}$

$\left\{\begin{array}{r} 7x - 2y = 4\\ -14x + 4y = -8 \end{array}\right.$

Exercise $\PageIndex{9}$

$\left\{\begin{array}{r} y = -x - 7\\ x = y - 5 \end{array}\right.$

Answer

$(-6, -1)$

Exercise $\PageIndex{10}$

$\left\{\begin{array}{r} 20x + 15y = -13\\ 5x - 20y = 13 \end{array}\right.$

Exercise $\PageIndex{11}$

$\left\{\begin{array}{r} x - 6y = 12\\ 4x + 6y = 18 \end{array}\right.$

Answer

$(6, -1)$

Exercise $\PageIndex{12}$

$\left\{\begin{array}{r} 8x + 9y = 0\\ 4x + 3y = 0 \end{array}\right.$

Exercise $\PageIndex{13}$

$\left\{\begin{array}{r} -5x + 2y = 1\\ 10x - 4y = -2 \end{array}\right.$

Answer

Dependent (same line)

Exercise $\PageIndex{14}$

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

Exercise $\PageIndex{15}$

$\left\{\begin{array}{r} 6x + 5y = 14\\ 4x - 8y = 32 \end{array}\right.$

Answer

$(4, -2)$

Exercise $\PageIndex{16}$

$\left\{\begin{array}{r} 5x - 7y = 4\\ 10x - 14y = 1 \end{array}\right.$

Exercise $\PageIndex{17}$

$\left\{\begin{array}{r} 2m + 10n = 0\\ -4m - 20n = -6 \end{array}\right.$

Answer

Inconsistent (parallel lines)

Exercise $\PageIndex{18}$

$\left\{\begin{array}{r} 7r - 2s = 6\\ -3r + 5s = -15 \end{array}\right.$

Exercise $\PageIndex{19}$

$\left\{\begin{array}{r} 28a - 21b = -19\\ 21a + 7b = 15 \end{array}\right.$

Answer

$(\dfrac{2}{7}, \dfrac{9}{7})$

Exercise $\PageIndex{20}$

$\left\{\begin{array}{r} 72x - 108y = 21\\ 18x + 36y = 25 \end{array}\right.$

Applications

Exercise $\PageIndex{21}$

The sum of two numbers is 35. One number is 7 larger than the other. What are the numbers?

Answer

The numbers are 14 and 21.

Exercise $\PageIndex{22}$

The difference of two numbers is 48. One number is three times larger than the other. What are the numbers?

Exercise $\PageIndex{23}$

A 35 pound mixture of two types of cardboard sells for $30.15. Type I cardboard sells for 90¢ a pound and type II cardboard sells for 75¢ a pound. How many pounds of each type of cardboard were used?

Answer

26 pounds at 90¢;  9 pounds at 75¢

Exercise $\PageIndex{24}$

The cost of 34 calculators of two different types is $1139. Type I calculator sells for $35 each and type II sells for $32 each. How many of each type of calculators were used?

Exercise $\PageIndex{25}$

A chemistry student needs 46 ml of a 15% salt solution. She has two salt solutions, A and B, to mix together to form the needed 46 ml solution. Salt solution A is 12% salt and salt solution B is 20% salt. How much of each solution should be used?

Answer

$28 \dfrac{3}{4}$ ml of solution A

$17 \dfrac{3}{4}$ ml of solution B

Exercise $\PageIndex{26}$

A chemist needs 100 ml of a 78% acid solution. He has two acid solutions to mix together to form the needed 100-ml solution. One solution is 50% acid and the other solution is 90% acid. How much of each solution should be used?

Exercise $\PageIndex{27}$

One third the sum of two numbers is 12 and half the difference is 14. What are the numbers?

Answer

$x=32, y=4$

Exercise $\PageIndex{28}$

Two angles are said to be complementary if their measures add to 90°. If one angle measures 8 more than four times the measure of its complement, find the measure of each of the angles.

Exercise $\PageIndex{29}$

A chemist needs 4 liters of a 20% acid solution. She has two solutions to mix together to form the 20% solution. One solution is 30% acid and the other solution is 24% acid. Can the chemist form the needed 20% acid solution? If the chemist locates a 14% acid solution, how much would have to be mixed with the 24% acid solution to obtain the needed 20% solution?

Answer

a) no solution

b) 1.6 liters (1600 ml) of the 14% solution

2.4 liters (2400 ml) of the 24% solution.

Exercise $\PageIndex{30}$

A chemist needs 80 ml of a 56% salt solution. She has a bottle of 60% salt solution. How much pure water and how much of the 60% salt solution should be mixed to dilute the 60% salt solution to a 56% salt solution?