$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 3.E: Solving Linear Systems

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Exercise $$\PageIndex{1}$$

Determine whether or not the given ordered pair is a solution to the given system.

1. $$\left( \frac { 2 } { 3 } , - 4 \right)$$;

$$\left\{ \begin{array} { l } { 9 x - y = 10 } \\ { 3 x + 4 y = - 14 } \end{array} \right.$$

2. $$\left( - \frac { 1 } { 2 } , \frac { 3 } { 4 } \right)$$;

$$\left\{ \begin{array} { l } { 6 x - 8 y = - 9 } \\ { x + 2 y = 1 } \end{array} \right.$$

3. $$\left( - 5 , - \frac { 7 } { 8 } \right)$$;

$$\left\{ \begin{array} { l } { x - 16 y = 9 } \\ { 2 x - 8 y = - 17 } \end{array} \right.$$

4. $$\left( - 1 , \frac { 4 } { 5 } \right)$$;

$$\left\{ \begin{array} { l } { 2 x + 5 y = 2 } \\ { 3 x - 10 y = - 5 } \end{array} \right.$$

1. Yes

3. No

Exercise $$\PageIndex{2}$$

Given the graphs, determine the simultaneous solution.

1.

2.

3.

4.

1. $$(-6, 2)$$

3. $$\varnothing$$

Exercise $$\PageIndex{3}$$

Solve by graphing.

1. $$\left\{ \begin{array} { l } { 2 x + y = 6 } \\ { x - 2 y = 8 } \end{array} \right.$$
2. $$\left\{ \begin{array} { l } { 5 x - 2 y = 0 } \\ { x - y = 3 } \end{array} \right.$$
3. $$\left\{ \begin{array} { l } { 4 x + 3 y = - 12 } \\ { - 8 x - 6 y = 24 } \end{array} \right.$$
4. $$\left\{ \begin{array} { l } { \frac { 1 } { 2 } x + 2 y = 6 } \\ { x + 4 y = - 1 } \end{array} \right.$$
5. $$\left\{ \begin{array} { l } { 5 x + 2 y = 30 } \\ { y - 5 = 0 } \end{array} \right.$$
6. $$\left\{ \begin{array} { l } { 5 x + 3 y = - 15 } \\ { x + 3 = 0 } \end{array} \right.$$
7. $$\left\{ \begin{array} { l } { \frac { 1 } { 3 } x - \frac { 1 } { 2 } y = 2 } \\ { \frac { 1 } { 2 } x + \frac { 3 } { 5 } y = 3 } \end{array} \right.$$
8. $$\left\{ \begin{array} { l } { \frac { 2 } { 5 } x + \frac { 1 } { 2 } y = 1 } \\ { \frac { 1 } { 15 } x + \frac { 1 } { 6 } y = - \frac { 1 } { 3 } } \end{array} \right.$$

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

3. $$\left( x , - \frac { 4 } { 3 } x - 4 \right)$$

5. $$(4,5)$$

7. $$(6,0)$$

Exercise $$\PageIndex{4}$$

Solve by substitution.

1. $$\left\{ \begin{array} { l } { 4 x - y = 12 } \\ { x + 3 y = - 10 } \end{array} \right.$$
2. $$\left\{ \begin{array} { l } { 9 x - 2 y = 3 } \\ { x - 3 y = 17 } \end{array} \right.$$
3. $$\left\{ \begin{array} { l } { 12 x + y = 7 } \\ { 3 x - 4 y = 6 } \end{array} \right.$$
4. $$\left\{ \begin{array} { l } { 3 x - 2 y = 1 } \\ { 2 x + 3 y = - 1 } \end{array} \right.$$

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

3. $$\left( \frac { 2 } { 3 } , - 1 \right)$$

Exercise $$\PageIndex{5}$$

Solve by elimination.

1. $$\left\{ \begin{array} { l } { 5 x - 2 y = - 12 } \\ { 4 x + 6 y = - 21 } \end{array} \right.$$
2. $$\left\{ \begin{array} { l } { 4 x - 5 y = 12 } \\ { 8 x + 3 y = - 2 } \end{array} \right.$$
3. $$\left\{ \begin{array} { l } { 5 x - 3 y = 11 } \\ { 2 x - 4 y = - 4 } \end{array} \right.$$
4. $$\left\{ \begin{array} { l } { 7 x + 2 y = 3 } \\ { 3 x + 5 y = - 7 } \end{array} \right.$$

1. $$\left( - 3 , - \frac { 3 } { 2 } \right)$$

3. $$(4,3)$$

Exercise $$\PageIndex{6}$$

Solve using any method.

1. $$\left\{ \begin{array} { l } { 4 x - 8 y = 4 } \\ { x + 2 y = 9 } \end{array} \right.$$
2. $$\left\{ \begin{array} { l } { 6 x - 9 y = 8 } \\ { x - y = 1 } \end{array} \right.$$
3. $$\left\{ \begin{array} { l } { 2 x - 6 y = - 1 } \\ { 6 x + 10 y = - 3 } \end{array} \right.$$
4. $$\left\{ \begin{array} { l } { 2 x - 3 y = 36 } \\ { x - 3 y = 9 } \end{array} \right.$$
5. $$\left\{ \begin{array} { l } { 5 x - 3 y = 10 } \\ { - 10 x + 6 y = 3 } \end{array} \right.$$
6. $$\left\{ \begin{array} { l } { \frac { 1 } { 2 } x - y = 3 } \\ { 3 x - 6 y = 18 } \end{array} \right.$$
7. $$\left\{ \begin{array} { l } { \frac { 3 } { 5 } x - \frac { 1 } { 2 } y = - 1 } \\ { \frac { 1 } { 10 } x + \frac { 3 } { 4 } y = - 1 } \end{array} \right.$$
8. $$\left\{ \begin{array} { l } { \frac { 4 } { 3 } x - \frac { 2 } { 5 } y = - \frac { 8 } { 15 } } \\ { \frac { 1 } { 2 } x - \frac { 2 } { 3 } y = - \frac { 11 } { 24 } } \end{array} \right.$$

1. $$(5,2)$$

3. $$\left( - \frac { 1 } { 2 } , 0 \right)$$

5. $$\varnothing$$

7. $$\left( - \frac { 5 } { 2 } , - 1 \right)$$

Exercise $$\PageIndex{7}$$

Set up a linear system and solve.

1. The sum of two integers is $$32$$. The larger is $$4$$ less than twice the smaller. Find the integers.
2. The sum of $$2$$ times a larger integer and $$3$$ times a smaller integer is $$54$$. When twice the smaller integer is subtracted from the larger, the result is $$−1$$. Find the integers.
3. The length of a rectangle is $$2$$ centimeters less than three times its width and the perimeter measures $$44$$ centimeters. Find the dimensions of the rectangle.
4. The width of a rectangle is one-third of its length. If the perimeter measures $$53 \frac{1}{3}$$ centimeters, then find the dimensions of the rectangle.
5. The sum of a larger integer and $$3$$ times a smaller is $$61$$. When twice the smaller integer is subtracted from the larger, the result is $$1$$. Find the integers.
6. A total of $$8,600$$ was invested in two accounts. One account earned $$4 \frac{3}{4}$$% annual interest and the other earned $$6 \frac{1}{2}$$% annual interest. If the total interest for one year was $$431.25$$, how much was invested in each account?
7. A jar consisting of only nickels and dimes contains $$76$$ coins. If the total value is $$6$$, how many of each coin are in the jar?
8. A nurse wishes to obtain $$32$$ ounces of a $$1.2$$% saline solution. How much of a $$1$$% saline solution must she mix with a $$2.6$$% saline solution to achieve the desired mixture?
9. A light aircraft flying with the wind can travel $$330$$ miles in $$2$$ hours. The aircraft can fly the same distance against the wind in $$3$$ hours. Find the speed of the wind.
10. An executive was able to average $$52$$ miles per hour to the airport in her car and then board an airplane that averaged $$340$$ miles per hour. If the total $$640$$-mile business trip took $$4$$ hours, how long did she spend on the airplane?

1. $$12, 20$$

3. Length: $$16$$ centimeters; width: $$6$$ centimeters

5. $$12, 25$$

7. The jar contains $$32$$ nickels and $$44$$ dimes.

9. $$27.5$$ miles per hour

Exercise $$\PageIndex{8}$$

Determine whether the given ordered triple is a solution to the given system.

1. $$(-2,-1,3)$$;

$$\left\{ \begin{array} { l } { 4 x - y + 2 z = - 1 } \\ { x - 4 y + 3 z = 11 } \\ { 3 x + 5 y - 4 z = 1 } \end{array} \right.$$

2. $$(5,-3,-2)$$;

$$\left\{ \begin{array} { l } { x - 4 y + 6 z = 5 } \\ { 2 x + 5 y - z = - 3 } \\ { 3 x - 4 y + z = 25 } \end{array} \right.$$

3. $$\left( 1 , - \frac { 3 } { 2 } , - \frac { 4 } { 3 } \right)$$;

$$\left\{ \begin{array} { l } { 5 x - 4 y + 3 z = 7 } \\ { x + 2 y - 6 z = 6 } \\ { 12 x - 6 y + 6 z = 13 } \end{array} \right.$$

4. $$\left( \frac { 5 } { 4 } , - \frac { 1 } { 3 } , 2 \right)$$;

$$\left\{ \begin{array} { c } { 8 x + 9 y + z = 9 } \\ { 4 x + 12 y - 4 z = - 7 } \\ { 12 x - 6 y - z = - 5 } \end{array} \right.$$

1. No

3. Yes

Exercise $$\PageIndex{9}$$

Solve.

1. $$\left\{ \begin{array} { c } { 2 x + 3 y - z = 1 } \\ { 5 y + 2 z = 12 } \\ { 3 z = 18 } \end{array} \right.$$
2. $$\left\{ \begin{array} { c } { 3 x - 5 y - 2 z = 21 } \\ { y - 7 z = 18 } \\ { 4 z = - 12 } \end{array} \right.$$
3. $$\left\{ \begin{array} { l } { 4 x - 5 y - z = - 6 } \\ { 3 x + 6 y + 5 z = 3 } \\ { 5 x - 2 y - 3 z = - 17 } \end{array} \right.$$
4. $$\left\{ \begin{array} { l } { x - 6 y + 3 z = - 2 } \\ { 5 x + 4 y - 2 z = 24 } \\ { 6 x - 8 y - 5 z = 25 } \end{array} \right.$$
5. $$\left\{ \begin{array} { l } { x + 2 y - 2 z = 1 } \\ { 2 x - y - z = - 2 } \\ { 6 x - 3 y - 3 z = 12 } \end{array} \right.$$
6. $$\left\{ \begin{array} { l } { 3 x + y + 2 z = - 1 } \\ { 9 x + 3 y + 6 z = - 3 } \\ { 4 x + y + 4 z = - 3 } \end{array} \right.$$
7. $$\left\{ \begin{array} { c } { 3 a - 2 b + 5 c = - 3 } \\ { 6 a + 4 b - c = - 2 } \\ { - 6 a + 6 b + 24 c = 7 } \end{array} \right.$$
8. $$\left\{ \begin{array} { c } { 9 a - 2 b - 6 c = 10 } \\ { 5 a - 3 b - 10 c = 14 } \\ { - 3 a + 4 b + 12 c = - 20 } \end{array} \right.$$

1. $$\left( \frac { 7 } { 2 } , 0,6 \right)$$

3. $$(-2,-1,3)$$

5. $$\varnothing$$

7. $$\left( - \frac { 2 } { 3 } , \frac { 1 } { 2 } , 0 \right)$$

Exercise $$\PageIndex{10}$$

Set up a linear system and solve.

1. The sum of three integers is $$24$$. The larger is equal to the sum of the two smaller integers. Three times the smaller is equal to the larger. Find the integers.
2. The sports center sold $$120$$ tickets to the Friday night basketball game for a total of $$942$$. A general admission ticket cost $$12$$, a student ticket cost $$6$$, and a child ticket cost $$4$$. If the sum of the general admission and student tickets totaled $$105$$, then how many of each ticket were sold?
3. A $$16$$-ounce mixed nut product containing $$13.5$$% peanuts is to be packaged. The packager has a three-mixed nut product containing $$6$$%, $$10$$%, and $$50$$% peanut concentrations in stock. If the amount of $$50$$% peanut product is to be one-quarter that of the $$10$$% peanut product, then how much of each will be needed to produce the desired peanut concentration?
4. Water is to be mixed with two acid solutions to produce a $$25$$-ounce solution containing $$6$$% acid. The acid mixtures on hand contain $$10$$% and $$25$$% acid. If the amount of $$25$$% acid is to be one-half the amount of the $$10$$% acid solution, how much water will be needed?

1. $$4,8,12$$

3. $$6$$ oz of the $$6$$% peanut stock, $$8$$ oz of the $$10$$% peanut stock, and $$2$$ oz of the $$50$$% peanut stock should be mixed.

Exercise $$\PageIndex{11}$$

Construct the corresponding augmented matrix.

1. $$\left\{ \begin{array} { l } { 9 x - 7 y = 4 } \\ { 3 x - y = - 1 } \end{array} \right.$$
2. $$\left\{ \begin{array} { c } { x - 5 y = 12 } \\ { 3 y = - 5 } \end{array} \right.$$
3. $$\left\{ \begin{array} { l } { x - y + 2 z = - 6 } \\ { 3 x - 6 y - z = 3 } \\ { - x + y - 5 z = 10 } \end{array} \right.$$
4. $$\left\{ \begin{array} { c } { 5 x + 7 y - z = 0 } \\ { - 8 y + z = - 1 } \\ { - x + 3 z = - 9 } \end{array} \right.$$

1. $$\left[ \begin{array} { r r | r } { 9 } & { - 7 } & { 4 } \\ { 3 } & { - 1 } & { - 1 } \end{array} \right]$$

3. $$\left[ \begin{array} { r r r | r } { 1 } & { - 1 } & { 2} &{- 6 } \\ { 3 } & { - 6 } & { - 1} & { 3 } \\ { - 1 } & { 1 } & { - 5 } & { 10 } \end{array} \right]$$

Exercise $$\PageIndex{12}$$

Solve using matrices and Gaussian elimination.

1. $$\left\{ \begin{array} { l } { 4 x + 5 y = 0 } \\ { 2 x - 3 y = 22 } \end{array} \right.$$
2. $$\left\{ \begin{array} { l } { 3 x - 8 y = 20 } \\ { 2 x + 5 y = 3 } \end{array} \right.$$
3. $$\left\{ \begin{array} { c } { x - y + 4 z = 1 } \\ { - 2 x + 3 y - 2 z = 0 } \\ { x - 6 y + 8 z = 8 } \end{array} \right.$$
4. $$\left\{ \begin{array} { l } { - x + 3 y - z = 1 } \\ { 3 x - 6 y + 2 z = - 4 } \\ { 4 x - 3 y + 2 z = - 7 } \end{array} \right.$$
5. $$\left\{ \begin{array} { l } { 5 x - 3 y - z = 2 } \\ { x - 6 y + z = 7 } \\ { 2 x + 6 y - 2 z = - 8 } \end{array} \right.$$
6. $$\left\{ \begin{array} { l } { x + 2 y + 3 z = 4 } \\ { x + 3 y + z = 3 } \\ { 2 x + 5 y + 4 z = 8 } \end{array} \right.$$
7. $$\left\{ \begin{array} { c } { 2 a + 5 b - c = 4 } \\ { 2 a + c = - 2 } \\ { a + b + 3 c = 6 } \end{array} \right.$$
8. $$\left\{ \begin{array} { c } { a + 2 b + 3 c = - 7 } \\ { 4 b - 2 c = 8 } \\ { 3 a - c = - 7 } \end{array} \right.$$

1. $$(5,-4)$$

3. $$\left( - 2 , - 1 , \frac { 1 } { 2 } \right)$$

5. $$\left( x , \frac { 2 } { 3 } x - 1,3 x + 1 \right)$$

7. $$(-2,2,2)$$

Exercise $$\PageIndex{13}$$

Calculate the determinant.

1. $$\left| \begin{array} { c c} { - 9}&{5 } \\ { - 1}&{3 } \end{array} \right|$$
2. $$\left| \begin{array} { c c } { - 5}&{5 } \\ { - 3}&{3 } \end{array} \right|$$
3. $$\left| \begin{array} { c c } { 0}&{7 } \\ { 2}&{3 } \end{array} \right|$$
4. $$\left| \begin{array} { l l } { 0 } & { b _ { 1 } } \\ { a _ { 2 } }&{b _ { 2 } } \end{array} \right|$$
5. $$\left| \begin{array} { r r r } { 2 } & { - 3 } & { 0 } \\ { 1 } & { - 2 } & { - 1 } \\ { 0 } & { 1 } & { 3 } \end{array} \right|$$
6. $$\left| \begin{array} { r r r } { 3 } & { 2 } & { - 1 } \\ { 1 } & { - 1 } & { 0 } \\ { 5 } & { - 2 } & { - 4 } \end{array} \right|$$
7. $$\left| \begin{array} { r r r } { 5 } & { - 3 } & { - 1 } \\ { 1 } & { - 6 } & { 1 } \\ { 2 } & { 6 } & { - 2 } \end{array} \right|$$
8. $$\left| \begin{array} { l l l } { a _ { 1 } } & { 0 } & { 0 } \\ { a _ { 2 } } & { b _ { 2 } } & { 0 } \\ { a _ { 3 } } & { b _ { 3 } } & { c _ { 3 } } \end{array} \right|$$

1. $$-22$$

3. $$-14$$

5. $$-1$$

7. $$0$$

Exercise $$\PageIndex{14}$$

Solve using Cramer's rule.

1. $$\left\{ \begin{array} { l } { 2 x - 3 y = - 4 } \\ { 3 x + 5 y = 1 } \end{array} \right.$$
2. $$\left\{ \begin{array} { c } { 3 x - y = 2 } \\ { - 2 x + 6 y = 1 } \end{array} \right.$$
3. $$\left\{ \begin{array} { l } { 3 x + 5 y = 6 } \\ { 6 x + y = - 6 } \end{array} \right.$$
4. $$\left\{ \begin{array} { l } { 6 x - 4 y = - 1 } \\ { - 3 x + 2 y = 2 } \end{array} \right.$$
5. $$\left\{ \begin{array} { l } { 5 x + 2 y + 4 z = 4 } \\ { 4 x + 3 y + 2 z = - 5 } \\ { - 5 x - 3 y - 5 z = 0 } \end{array} \right.$$
6. $$\left\{ \begin{array} { l } { 2 x - y + 2 z = 1 } \\ { x - 3 y + z = 2 } \\ { 3 x - y - 4 z = - 2 } \end{array} \right.$$
7. $$\left\{ \begin{array} { l } { 4 x - y - 2 z = - 7 } \\ { 2 x + y + 6 z = 0 } \\ { 2 x + 2 y + 4 z = - 1 } \end{array} \right.$$
8. $$\left\{ \begin{array} { c } { x - y - z = 1 } \\ { 2 x - y + 3 z = 2 } \\ { x + y + z = - 1 } \end{array} \right.$$
9. $$\left\{ \begin{array} { l } { 4 x - y + 2 z = - 1 } \\ { 2 x + 3 y - z = 3 } \\ { 6 x + 2 y + z = 2 } \end{array} \right.$$
10. $$\left\{ \begin{array} { l } { x - y + 2 z = 1 } \\ { 2 x + 2 y - z = 2 } \\ { 3 x + y + z = 1 } \end{array} \right.$$

1. $$\left( - \frac { 17 } { 19 } , \frac { 14 } { 19 } \right)$$

3. $$\left( - \frac { 4 } { 3 } , 2 \right)$$

5. $$(2,-5,1)$$

7. $$\left( - \frac { 3 } { 2 } , 0 , \frac { 1 } { 2 } \right)$$

9. $$\left( x , - \frac { 8 } { 5 } x + 1 , - \frac { 14 } { 5 } x \right)$$

Exercise $$\PageIndex{15}$$

Determine whether or not the given point is a solution to the system of inequalities.

1. $$(-6,1)$$;

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

2. $$\left( \frac { 1 } { 2 } , - 3 \right)$$;

$$\left\{ \begin{array} { l } { 4 x - 2 y \geq 8 } \\ { 6 x + 2 y < - 3 } \end{array} \right.$$

3. $$(-4,-2)$$;

$$\left\{ \begin{array} { l } { x - y > - 3 } \\ { 2 x + 3 y \leq 0 } \\ { - 3 x + 4 y \geq 4 } \end{array} \right.$$

4. $$\left( 5 , - \frac { 1 } { 5 } \right)$$'

$$\left\{ \begin{array} { l } { y < x ^ { 2 } - 25 } \\ { y > \frac { 2 } { 3 } x - 1 } \end{array} \right.$$

5. $$(-3,-2)$$;

$$\left\{ \begin{array} { l } { y < ( x - 1 ) ^ { 2 } } \\ { y \leq | x + 1 | - 3 } \end{array} \right.$$

6. $$\left( 2 , - \frac { 2 } { 3 } \right)$$;

$$\left\{ \begin{array} { l } { y < 0 } \\ { x ^ { 2 } + y \geq 3 } \end{array} \right.$$

1. Yes

3. Yes

5. Yes

Exercise $$\PageIndex{16}$$

Graph the solution set.

1. $$\left\{ \begin{array} { l } { y \leq - 4 } \\ { x - 2 y > 8 } \end{array} \right.$$
2. $$\left\{ \begin{array} { l } { x + 4 y > 8 } \\ { 2 x - y \leq 4 } \end{array} \right.$$
3. $$\left\{ \begin{array} { l } { y - 3 < 0 } \\ { - 2 x + 3 y > - 9 } \\ { x + y \geq 1 } \end{array} \right.$$
4. $$\left\{ \begin{array} { l } { y \leq 0 } \\ { 2 x - 6 y < 9 } \\ { - 2 x + 6 y < 9 } \end{array} \right.$$
5. $$\left\{ \begin{array} { l } { 2 x + y < 3 } \\ { y > ( x - 2 ) ^ { 2 } - 5 } \end{array} \right.$$
6. $$\left\{ \begin{array} { l } { y > | x | } \\ { y \geq - x ^ { 2 } + 6 } \end{array} \right.$$
7. $$\left\{ \begin{array} { l } { x - 2 y < 12 } \\ { y \leq ( x - 4 ) ^ { 3 } } \end{array} \right.$$
8. $$\left\{ \begin{array} { l } { y + 6 > 0 } \\ { y < \sqrt { x } } \end{array} \right.$$

1.

3.

5.

7.

## Sample Exam

Exercise $$\PageIndex{17}$$

1. Determine whether or not $$(-2, \frac{3}{4})$$ is a solution to $$\left\{ \begin{array} { l } { 2 x - 8 y = - 10 } \\ { 3 x + 4 y = - 3 } \end{array} \right.$$.
2. Determine whether or not $$(−3, 2, −5)$$ is a solution to $$\left\{ \begin{array} { l } { x - y + 2 z = - 15 } \\ { 2 x - 3 y + z = - 17 } \\ { 3 x + 5 y - 2 z = 10 } \end{array} \right.$$.

1. Yes

Exercise $$\PageIndex{18}$$

Solve by graphing.

1. $$\left\{ \begin{array} { l } { x - y = - 5 } \\ { x + y = - 3 } \end{array} \right.$$
2. $$\left\{ \begin{array} { l } { 6 x - 8 y = 48 } \\ { \frac { 1 } { 2 } x - \frac { 2 } { 3 } y = 1 } \end{array} \right.$$
3. $$\left\{ \begin{array} { l } { \frac { 1 } { 2 } x + y = - 6 } \\ { - 2 x - 4 y = 24 } \end{array} \right.$$

1. $$(-4,1)$$

3. $$\left( x , - \frac { 1 } { 2 } x - 6 \right)$$

Exercise $$\PageIndex{19}$$

Solve by substitution.

1. $$\left\{ \begin{array} { l } { x - 8 y = 10 } \\ { 3 x + 2 y = 17 } \end{array} \right.$$
2. $$\left\{ \begin{array} { l } { \frac { 3 } { 2 } x - \frac { 1 } { 6 } y = - \frac { 23 } { 2 } } \\ { \frac { 3 } { 8 } x + \frac { 5 } { 6 } y = - \frac { 11 } { 2 } } \end{array} \right.$$
3. $$\left\{ \begin{array} { l } { 5 x - y = 15 } \\ { 2 x - \frac { 2 } { 5 } y = 6 } \end{array} \right.$$

2. $$(-8,-3)$$

Exercise $$\PageIndex{20}$$

Solve.

1. $$\left\{ \begin{array} { l } { 3 x - 5 y = 27 } \\ { 7 x + 2 y = 22 } \end{array} \right.$$
2. $$\left\{ \begin{array} { c } { 12 x + 3 y = - 3 } \\ { 5 x + 2 y = 1 } \end{array} \right.$$
3. $$\left\{ \begin{array} { l } { 5 x - 3 y = - 1 } \\ { - 15 x + 9 y = 5 } \end{array} \right.$$
4. $$\left\{ \begin{array} { l } { 6 a - 3 b + 2 c = 11 } \\ { 2 a - b - 4 c = - 15 } \\ { 4 a - 5 b + 3 c = 23 } \end{array} \right.$$
5. $$\left\{ \begin{array} { l } { 4 x + y - 6 z = 8 } \\ { 5 x + 4 y - 2 z = 10 } \\ { 2 x + y - 2 z = 4 } \end{array} \right.$$

1. $$(4,-3)$$

3. $$\varnothing$$

5. $$\left( x , - x + 2 , \frac { 1 } { 2 } x - 1 \right)$$

Exercise $$\PageIndex{21}$$

Solve using any method.

1. $$\left\{ \begin{array} { c } { x - 5 y + 8 z = 1 } \\ { 2 x + 9 y - 4 z = - 8 } \\ { - 3 x + 11 y + 12 z = 15 } \end{array} \right.$$
2. $$\left\{ \begin{array} { l } { 2 x - y + z = 1 } \\ { x - y + 3 z = 2 } \\ { 3 x - 2 y + 4 z = 5 } \end{array} \right.$$
3. $$\left\{ \begin{array} { l } { - 5 x + 3 y = 2 } \\ { 4 x + 2 y = - 1 } \end{array} \right.$$
4. $$\left\{ \begin{array} { l } { 2 x - 3 y + 2 z = 2 } \\ { x + 2 y - 3 z = 0 } \\ { - x - y + z = - 2 } \end{array} \right.$$

2. $$\varnothing$$

4. $$(2,2,2)$$

Exercise $$\PageIndex{22}$$

Graph the solution set.

1. $$\left\{ \begin{array} { l } { 3 x + 4 y < 24 } \\ { 2 x - 3 y \leq 3 } \\ { y + 1 > 0 } \end{array} \right.$$
2. $$\left\{ \begin{array} { l } { x + y < 4 } \\ { y > - ( x + 6 ) ^ { 2 } + 4 } \end{array} \right.$$

2.

Exercise $$\PageIndex{23}$$

Use algebra to solve the following.

1. The length of a rectangle is $$1$$ inch less than twice that of its width. If the perimeter measures $$49$$ inches, then find the dimensions of the rectangle.
2. Joe’s $$4,000$$ savings is in two accounts. One account earns $$3.1$$% annual interest and the other earns $$4.9$$% annual interest. His total interest for the year is $$174.40$$. How much does he have in each account?
3. One solution contains $$40$$% alcohol and another contains $$72$$% alcohol. How much of each should be mixed together to obtain $$16$$ ounces of a $$62$$% alcohol solution?
4. Jerry took two buses on the $$193$$-mile trip to visit his grandmother. The first bus averaged $$46$$ miles per hour and the second bus was able to average $$52$$ miles per hour. If the total trip took $$4$$ hours, then how long was spent in each bus?
5. A total of $$8,500$$ was invested in three interest earning accounts. The interest rates were $$2$$%, $$3$$%, and $$6$$%. If the total simple interest for one year was $$380$$ and the amount invested at $$6$$% was equal to the sum of the amounts in the other two accounts, then how much was invested in each account?
6. A mechanic wishes to mix $$6$$ gallons of a $$22$$% antifreeze solution. In stock he has a $$60$$% and an $$80$$% antifreeze concentrate. Water is to be added in the amount that is equal to twice the amount of both concentrates combined. How much water is needed?
2. Joe has $$1,200$$ in the account earning $$3.1$$% interest and $$2,800$$ in the account earning $$4.9$$% interest
4. Jerry spent $$2.5$$ hours in the first bus and $$1.5$$ hours in the second.
6. $$4$$ gallons of water is needed.