Question

CRITICAL THINKING Determine values for each variable if $d=1$, $e=4d$, $z+d=e$, $f=\frac{x}{5}$, $ay=1.5$, $x=\frac{d}{2}$and $y=x+\frac{x}{2}$.

$a\left[\begin{array}{ccc}x& y& z\\ d& e& f\end{array}\right]=\left[\begin{array}{ccc}ax& ay& az\\ ad& ae& af\end{array}\right]$

Short answer

The values for each variable $a,d,e,f,x,y\text{and}z$ are 2, 1, 4, $\frac{1}{10}$, $\frac{1}{2}$, $\frac{3}{4}$, and 3 respectively.

Step-by-step solution

- Solve for variables e and x

Substitute 1 for d into the equations $e=4d$and $x=\frac{d}{2}$ and solve for e and x.

$\begin{array}{c}e=4\left(1\right)\\ =4\end{array}$

And,

$\begin{array}{c}x=\frac{\left(1\right)}{2}\\ =\frac{1}{2}\end{array}$

- Solve for variables f and y

Substitute $\frac{1}{2}$ for x into the equations $f=\frac{x}{5}$and $y=x+\frac{x}{2}$ and solve for f and y.

$\begin{array}{c}f=\frac{\left(\frac{1}{2}\right)}{5}\\ =\frac{1}{10}\end{array}$

And,

$\begin{array}{c}y=\frac{1}{2}+\frac{\left(\frac{1}{2}\right)}{2}\\ =\frac{1}{2}+\frac{1}{4}\\ =\frac{3}{4}\end{array}$

- Solve for variable z and a

Substitute 1 for d and 4 for e into the equation $z+d=e$ and solve for z.

$\begin{array}{c}z+\left(1\right)=\left(4\right)\\ z+1-1=4-1\\ z=3\end{array}$

Now substitute $\frac{3}{4}$for $y$into the equation $ay=1.5$and solve for a.

$\begin{array}{c}a\left(\frac{3}{4}\right)=1.5\\ a=1.5\times \frac{4}{3}\\ a=2\end{array}$