Question

tell whether each equation is a direct variation. if so, identify the constant of variation. (lesson 5-5)

1. ( x = 2y )
2. ( y = -6x )
3. ( y - 1 = x )

solve each system by substitution. (lesson 6-2)

1. ( \begin{cases} y = x - 1 \\ x + y = 10 end{cases} )
2. ( \begin{cases} x = y - 5 \\ 2x + 1 = y end{cases} )
3. ( \begin{cases} y = 2x - 1 \\ x - y = 3 end{cases} )

Explanation:

First Section: Direct Variation Check

Problem 43: $x=2y$

Step1: Rewrite to $y=kx$ form

$y=\frac{1}{2}x$

Step2: Identify direct variation

Matches $y=kx$, $k=\frac{1}{2}$

Problem 44: $y=-6x$

Step1: Match to $y=kx$ form

Already in $y=kx$ format

Step2: Identify constant $k$

$k=-6$

Problem 45: $y-1=x$

Step1: Rewrite to $y=mx+b$ form

$y=x+1$

Step2: Check direct variation

Has a constant term, not $y=kx$

Problem 46:

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

Step1: Substitute $y=x-1$ into 2nd eq

$x+(x-1)=10$

Step2: Simplify and solve for $x$

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

Step3: Substitute $x$ back to find $y$

$y=5.5-1=4.5=\frac{9}{2}$

Problem 47:

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

Step1: Substitute $x=y-5$ into 2nd eq

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

Step2: Simplify and solve for $y$

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

Step3: Substitute $y$ back to find $x$

$x=9-5=4$

Problem 48:

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

Step1: Substitute $y=2x-1$ into 2nd eq

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

Step2: Simplify and solve for $x$

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

Step3: Substitute $x$ back to find $y$

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

Answer:

1. This is direct variation, constant of variation is $\frac{1}{2}$
2. This is direct variation, constant of variation is $-6$
3. This is not a direct variation

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Second Section: Solve Systems by Substitution