Question

1. $-\frac{4}{9}x + \frac{1}{3}y = \frac{8}{9}$

$\frac{4}{3}x - \frac{7}{6}y = -2$

1. $-\frac{3}{8}x + \frac{5}{8}y = \frac{1}{4}$

$-\frac{3}{5}x + \frac{9}{10}y = \frac{3}{5}$

1. $\

$$\begin{cases}1.6x - 1.8y = -2.2 \\\\-0.8x + y = 1.4\\end{cases}$$

$

1. $\

$$\begin{cases}-1.4x - 0.6y = -1.8 \\\\-1.2x - 0.4y = -2\\end{cases}$$

$

Explanation:

Problem 9

Step1: Eliminate denominators (Eq1)

Multiply Eq1 by 9:
$$-\frac{4}{9}x \times 9 + \frac{1}{3}y \times 9 = \frac{8}{9} \times 9$$
$$-4x + 3y = 8 \tag{1a}$$

Step2: Eliminate denominators (Eq2)

Multiply Eq2 by 6:
$$\frac{4}{3}x \times 6 - \frac{7}{6}y \times 6 = -2 \times 6$$
$$8x - 7y = -12 \tag{2a}$$

Step3: Scale Eq1a to eliminate x

Multiply (1a) by 2:
$$-8x + 6y = 16 \tag{1b}$$

Step4: Add (1b) and (2a)

$$(-8x + 6y) + (8x - 7y) = 16 + (-12)$$
$$-y = 4 \implies y = -4$$

Step5: Substitute y=-4 into (1a)

$$-4x + 3(-4) = 8$$
$$-4x -12 = 8$$
$$-4x = 20 \implies x = -5$$

Step1: Eliminate denominators (Eq1)

Multiply Eq1 by 8:
$$-\frac{3}{8}x \times 8 + \frac{5}{8}y \times 8 = \frac{1}{4} \times 8$$
$$-3x + 5y = 2 \tag{1a}$$

Step2: Eliminate denominators (Eq2)

Multiply Eq2 by 10:
$$-\frac{3}{5}x \times 10 + \frac{9}{10}y \times 10 = \frac{3}{5} \times 10$$
$$-6x + 9y = 6 \tag{2a}$$

Step3: Scale Eq1a to eliminate x

Multiply (1a) by 2:
$$-6x + 10y = 4 \tag{1b}$$

Step4: Subtract (2a) from (1b)

$$(-6x + 10y) - (-6x + 9y) = 4 - 6$$
$$y = -2$$

Step5: Substitute y=-2 into (1a)

$$-3x + 5(-2) = 2$$
$$-3x -10 = 2$$
$$-3x = 12 \implies x = -4$$

Step1: Eliminate decimals (Eq1)

Multiply Eq1 by 10:
$$1.6x \times 10 - 1.8y \times 10 = -2.2 \times 10$$
$$16x - 18y = -22 \tag{1a}$$

Step2: Eliminate decimals (Eq2)

Multiply Eq2 by 10:
$$-0.8x \times 10 + y \times 10 = 1.4 \times 10$$
$$-8x + 10y = 14 \tag{2a}$$

Step3: Scale Eq2a to eliminate x

Multiply (2a) by 2:
$$-16x + 20y = 28 \tag{2b}$$

Step4: Add (1a) and (2b)

$$(16x - 18y) + (-16x + 20y) = -22 + 28$$
$$2y = 6 \implies y = 3$$

Step5: Substitute y=3 into (2a)

$$-8x + 10(3) = 14$$
$$-8x + 30 = 14$$
$$-8x = -16 \implies x = 2$$

Answer:

$x=-5,\ y=-4$

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Problem 10