4.7: Systems of Equations- Answers to the Homework Exercises

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Graphing

$(-1,2)$

No Solution

$(-3,-1)$

$(1,-3)$

$(2,-2)$

$(3,2)$

$(-1,-2)$

$(4,-2)$

$(-3,1)$

$(4,4)$

$(4,-4)$

No Solution

$(2,-1)$

$(2,3)$

$(-3,1)$

The Substitution Method

$(1, −3)$; consistent; independent

$(−1, −2)$; consistent; independent

$(3, 3)$; consistent; independent

$(−2, −6)$; consistent; independent

$(−1, 8)$; consistent; independent

$(1, 7)$; consistent; independent

$(1, 3)$; consistent; independent

$(4, −3)$; consistent; independent

$(0, 3)$; consistent; independent

$(−4, −1)$; consistent; independent

$(−3, 3)$; consistent; independent

$(−1, 0)$; consistent; independent

$(8, −8)$; consistent; independent

$(1, −3)$; consistent; independent

$(−4, 3)$; consistent; independent

$(0, 2)$; consistent; independent

$(4, −2)$; consistent; independent

$(−4, 8)$; consistent; independent

$(1, −4)$; consistent; independent

$(2, 5)$; consistent; independent

The Addition Method

$(−2, 4)$; consistent; independent

No solution; inconsistent

$(−3, −5)$; consistent; independent

$(0, 4)$; consistent; independent

$(4, 6)$; consistent; independent

$(0, −4)$; consistent; independent

$(−1, −2)$; consistent; independent

$(−8, 9)$; consistent; independent

$(0, 0)$; consistent; independent

Infinite number of solutions; consistent; dependent

$(2, −2)$; consistent; independent

$(1, −2)$; consistent; independent

$(0, 3)$; consistent; independent

$(1, 2)$; consistent; independent

$(2, −2)$; consistent; independent

$(−3, 0)$; consistent; independent

$(−1, 1)$; consistent; independent

Applications with Systems of Equations

$236$ adult, $342$ child

$130$ adults, $70$ students

$73$ hotdogs, $58$ hamburgers

$$12,500 \:@\: 12\%$; $$14,500\: @\: 13\%$

$$2,500\: @\: 10\%$; $$6,500\: @\: 12\%$

$$4,100\: @\: 9.5\%$; $$5,900\: @\: 11\%$

$$1,600\: @\: 4\%$; $$2,400\: @\: 8\%$

$$3,500\: @\: 6\%$; $$5,000\: @\: 3.5\%$

$$6,500\: @\: 8\%$; $$8,500\: @\: 11\%$

$$3,000\: @\: 4.25\%$; $$3,000\: @\: 5.75\%$

$$7,500\: @\: 6.8\%$; $$3,500\: @\: 8.2\%$

$$5,000\: @\: 12\%$; $$11,000\: @\: 8\%$

$30,\: 70$

$40,\: 110$

$100,\: 200$

$10,\: 5$

$21,\: 49$

$2,\: 3$

$1.5,\: 3.5$

$55,\: 20$

$35,\: 63$

$1.2$

$10$

$75$

$63$ mph in still water; $9$ mph current’s rate

$5$ mph in still water; $1$ mph current’s rate

$\frac{783}{8}$ mph in still air; $\frac{113}{8}$ mph wind’s rate

$35$ mph in still air; $5$ mph wind’s rate

Systems of Three Linear Equations in Three Variables

$(1, −1, 2)$; consistent; independent

$(−2, −1, 4)$; consistent; independent

$(0, 0, 0)$; consistent; independent

$(0, 0, 0)$; consistent; independent

$(1, 2, 3)$; consistent; independent

$(10, 2, 3)$; consistent; independent

$(3, −2, 1)$; consistent; independent

$\{(x, y, z)|x + y − z = 0\}$; consistent; dependent

$\{(x, y, z)|4x + 12y + 16z = 4\}$; consistent; dependent

$\{(p, q, r)|p + q + r = 1\}$; consistent; dependent

no solution; inconsistent

$(2, 3, 1)$; consistent; independent

$(−25, 18, −25)$; consistent; independent

$(1, 2, 4)$; consistent; independent

$(1, −3, −2, −1)$; consistent; independent

$(7, 4, 5, 6)$; consistent; independent

Systems of Two Linear Inequalities in Two Variables

yes, no, yes

yes, no, no

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