Question

what is the solution to this system of equations?
y = 9x + 2
y = 3x + 2
a) (-2, 0) c) (0, 2)
b) (3, 7) d) (-7, -3)

⑦ what is the solution to this system of equations?
3a - b + 14c = 7
2a + 2b + 3c = 0
a - 12b - 18c = 33

⑧ which equation in slope - intercept form best represents the line that passes through these two points?
(1, -4)
(-2, 6)
a) $y = -\frac{2}{3}x - \frac{11}{3}$ c) $y = -\frac{2}{3}x - \frac{10}{3}$
b) $y = -\frac{11}{3}x - \frac{2}{3}$ d) $y = -\frac{10}{3}x - \frac{2}{3}$

⑨ which equation in slope - intercept form best represents the line that passes through these two points?
(0, 4)
(2, 0)
a) $y = 4x + 2$ c) $y = -2x + 0$
b) $y = -2x + 4$ d) $y = 4x + 0$

Explanation:

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Problem 1

Step1: Set equations equal

$9x + 2 = 3x + 2$

Step2: Solve for x

$9x - 3x = 2 - 2$
$6x = 0$
$x = 0$

Step3: Find y value

$y = 9(0) + 2 = 2$
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Problem 2

Step1: Eliminate b (Eq1×2+Eq2)

$2(3a - b + 14c) + (2a + 2b + 3c) = 2(7) + 0$
$6a - 2b + 28c + 2a + 2b + 3c = 14$
$8a + 31c = 14$ (Eq4)

Step2: Eliminate b (Eq1×12-Eq3)

$12(3a - b + 14c) - (a - 12b - 18c) = 12(7) - 33$
$36a - 12b + 168c - a + 12b + 18c = 84 - 33$
$35a + 186c = 51$ (Eq5)

Step3: Solve for c (Eq4×35-Eq5×8)

$35(8a + 31c) - 8(35a + 186c) = 35(14) - 8(51)$
$280a + 1085c - 280a - 1488c = 490 - 408$
$-403c = 82$
$c = -2$

Step4: Solve for a (sub c to Eq4)

$8a + 31(-2) = 14$
$8a - 62 = 14$
$8a = 76$
$a = 5$

Step5: Solve for b (sub a,c to Eq1)

$3(5) - b + 14(-2) = 7$
$15 - b - 28 = 7$
$-b -13 = 7$
$-b = 20$
$b = -4$
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Problem 3

Step1: Calculate slope

$m = \frac{6 - (-4)}{-2 - 1} = \frac{10}{-3} = -\frac{2}{3}$

Step2: Find y-intercept (sub (1,-4))

$-4 = -\frac{2}{3}(1) + b$
$-4 + \frac{2}{3} = b$
$b = -\frac{10}{3}$

Step3: Write slope-intercept form

$y = -\frac{2}{3}x - \frac{10}{3}$
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Problem 4

Step1: Calculate slope

$m = \frac{0 - 4}{2 - 0} = \frac{-4}{2} = -2$

Step2: Identify y-intercept

From point (0,4), $b=4$

Step3: Write slope-intercept form

$y = -2x + 4$

Answer:

1. c) (0,2)
2. $a=5$, $b=-4$, $c=-2$
3. c) $y=-\frac{2}{3}x-\frac{10}{3}$
4. b) $y=-2x+4$