Chapter 5.2, Problem 23P

Suppose $y$ varies directly with $x$.

Write and graph a direct variation equation that relates $x$ and $y$. $y=\frac{5}{8}$

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

To determine

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

Answer to Problem 23P

The direct variation equation that relations x and y is $\boxed{y=\frac{5}{4}x}$

Explanation of Solution

Given information : $y=\frac{5}{8}$ when $x=\frac{1}{2}$

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 $y=\frac{5}{8}$ & $x=\frac{1}{2}$ in to the equation

$\frac{5}{8}=k\left(\frac{1}{2}\right)$

Solve for k

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

$\frac{1}{2}k=\frac{5}{8}$

To get k alone multiply by 2 on both sides of the equation

$2\cdot \frac{1}{2}k=2\cdot \frac{5}{8}$

Simplify fraction in left side and rewrite the whole number as a fraction in right side $k=\frac{2}{1}\cdot \frac{5}{8}$

Multiply numerator with numerator and denominator with denominator in right side of the equation

$k=\frac{10}{8}$

Divide the numerator and denominator by common factor of the fraction

  $k=\frac{10÷2}{8÷2}$

Simplify fraction in right side of the equation

$k=\frac{5}{4}$

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

$y=\frac{5}{4}x$

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.

$x$	$y=\frac{5}{4}x$	$\left(x,y\right)$
$-4$	$\frac{5}{4}\cdot -4=-5$	$\left(-4,-5\right)$
$0$	$\frac{5}{4}\cdot 0=0$	$\left(0,0\right)$
$4$	$\frac{5}{4}\cdot 4=5$	$\left(4,5\right)$

Graph:

Conclusion:

The direct variation equation that relations x and y is $\boxed{y=\frac{5}{4}x}$