Question

spiral review determine a relationship between the x- and y-values. write an equation. (lesson 4-3) 40.

x	1	2	3	4
y	6	7	8	9

41.

x	1	2	3	4
y	3	6	9	12

42.

x	1	2	3	4
y	-9	-8	-7	-6

tell whether each equation is a direct variation. if so, identify the constant of variation. (lesson 5-5) 43. $x = 2y$ 44. $y = -6x$ 45. $y - 1 = x$ solve each system by substitution. (lesson 6-2) 46. $\

$$\begin{cases} y = x - 1 \\\\ x + y = 10 \\end{cases}$$

$ 47. $\

$$\begin{cases} x = y - 5 \\\\ 2x + 1 = y \\end{cases}$$

$ 48. $\

$$\begin{cases} y = 2x - 1 \\\\ x - y = 3 \\end{cases}$$

$ 6-3 solving systems by elimination

Explanation:

Questions 40-42: Determine x-y relationship, write equation

40. Step1: Find y-x difference

$6-1=5$, $7-2=5$, $8-3=5$, $9-4=5$

40. Step2: Write linear equation

$y = x + 5$
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41. Step1: Find y/x ratio

$\frac{3}{1}=3$, $\frac{6}{2}=3$, $\frac{9}{3}=3$, $\frac{12}{4}=3$

41. Step2: Write proportional equation

$y = 3x$
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42. Step1: Find y+x sum

$-9+1=-8$, $-8+2=-6$, $-7+3=-4$, $-6+4=-2$ (or find y-x difference: $-9-1=-10$, $-8-2=-10$, $-7-3=-10$, $-6-4=-10$)

42. Step2: Write linear equation

$y = x - 10$

A direct variation has the form $y=kx$ (k is constant, no constant term). Rearrange each equation to check this form.

1. For $x=2y$, rearrange to $y=\frac{1}{2}x$: fits $y=kx$.
2. For $y=-6x$: directly fits $y=kx$.
3. For $y-1=x$, rearrange to $y=x+1$: has a constant term, so not direct variation.

46. Step1: Substitute $y=x-1$ into second equation

$x + (x - 1) = 10$

46. Step2: Solve for x

$2x - 1 = 10 \implies 2x=11 \implies x=\frac{11}{2}=5.5$

46. Step3: Solve for y

$y = 5.5 - 1 = 4.5$
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47. Step1: Substitute $x=y-5$ into second equation

$2(y - 5) + 1 = y$

47. Step2: Simplify and solve for y

$2y - 10 + 1 = y \implies 2y - y = 9 \implies y=9$

47. Step3: Solve for x

$x = 9 - 5 = 4$
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48. Step1: Substitute $y=2x-1$ into second equation

$x - (2x - 1) = 3$

48. Step2: Simplify and solve for x

$x - 2x + 1 = 3 \implies -x=2 \implies x=-2$

48. Step3: Solve for y

$y = 2(-2) - 1 = -5$

Answer:

1. $y = x + 5$
2. $y = 3x$
3. $y = x - 10$

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Questions 43-45: Identify direct variation