Chapter 5.2, Problem 21P

Suppose $y$ varies directly with x. Write and graph a direct variation equation that relates $x$ and $y$. $y=-6$

when $x=3$

To determine

To find : direct variation equation and its graph that relates x and y

Answer to Problem 21P

The direct variation equation that relations x and y is $\boxed{y=-2x}$

Explanation of Solution

Given information : $y=-6$ when $x=3$

Concept Involved:

Use the fact that if “ y varies directly with x ” then the equation relating x and y is $y=kx$ where $k$ is constant of proportionality to set up an equation. Substitute the given values of x and y in to the equation. Solve the equation for k . Back Substitute the k value in equation $y=kx$ to give the relationship between x and y . To graph the equation, identify any two points satisfying the equation by picking some x values substituting in equation and finding the corresponding y value.

Calculation: Use the fact that if “ y varies directly with x ” then the equation relating x and y is $y=kx$ where $k$ is constant of proportionality to set up an equation.

$y=kx$

Substitute $y=-6$ & $x=3$ in to the equation

$-6=k\left(3\right)$

Solve for k

Use the symmetric property of identity which states that if a=b, then b=a to rewrite the equation

$3k=-6$

To get k alone divide by 3 on both sides of the equation

$\frac{3k}{3}=\frac{-6}{3}$

Simplify fraction on both sides

$k=-2$

Back Substitute the k value in equation $y=kx$ to give the relationship between x and y .

$y=-2x$

To graph the equation, identify any two points satisfying the equation by picking some x values substituting in equation and finding the corresponding y value.

$x$	$y=-2x$	$\left(x,y\right)$
$-2$	$-2\cdot -2=4$	$\left(-2,4\right)$
$0$	$-2\cdot 0=0$	$\left(0,0\right)$
$2$	$-2\cdot 2=-4$	$\left(2,-4\right)$

Graph:

Conclusion:

The direct variation equation that relations x and y is $\boxed{y=-2x}$