The following system of equations needs to be solved. (Find an approximate solution graphically, then refine this with appropriate calculations to an accuracy of 3 decimal places.

$$
\begin{gathered}
x+2 \sqrt{y}=6 \\
\sqrt{x}+y=4
\end{gathered}
$$
The desired ranges for $x$ and $y$ values in the graphs are limited by the requirement that neither under the square root nor the result of the square root can be a negative number, hence

$$
0 \leq x \leq 6 \quad \text { and } \quad 0 \leq y \leq 4
$$

From (1) and (2), we get

$$
\begin{gathered}
y=f(x)=\frac{(6-x)^{2}}{4} \\
y=g(x)=4-\sqrt{x}
\end{gathered}
$$

The graphs of these functions are depicted in part (a) of the figure, and they intersect near $x=3, y=2.2$. (No other intersection points are expected, given our familiarity with the shape of a parabola from practical graphing experience.)
![](https://cdn.mathpix.com/cropped/2024_05_02_0fba765381b4b6fadd49g-1.jpg?height=594&width=1026&top_left_y=703&top_left_x=539)

Indeed, (2) can also be written in the form $x=(4-y)^{2}$, so this graph is a part of a parabola with a "horizontal" axis. At $x=3$, the difference in the ordinates of the curves $f$ and $g$ (using square and square root tables to three decimal places):

$$
f(3)-g(3)=2.25-2.268=-0.018
$$

is negative, indicating that the curve of $f$ is below the curve of $g$ at this point, so the intersection point must have $x<3$. Let's try a smaller value, $x=2.95$:

$$
f(2.95)-g(2.95)=2.326-2.282=0.044
$$

is positive, indicating that the curve of $f$ is still above the curve of $g$ at this point, so the root must be greater than 2.95. Combining this with the previous result, we have $2.95 < x < 3$.

We draw an enlarged diagram of this segment (part (b) of the figure) based on our two points. The intersection of the chords of the curves is slightly to the right of the $x=2.985$ grid line, so let's try this value. The square root, rounded to three decimal places, is 1.727, with a maximum error of 0.001. Using an iterative method known from the textbook, we get a more precise approximation:

$$
\frac{1}{2}\left(1.727+\frac{2.985}{1.727}\right)=1.727715
$$

so $g(2.985)=2.272285$. On the other hand, by multiplication, rounded to six decimal places, $f(2.985)=2.272556$, which is 0.000271 more than $g(2.985)$, indicating that the root is still greater than 2.985.

Repeating this calculation:

$$
f(2.986)-g(2.986)=2.271049-2.271995=-0.000946<0
$$

so the root is between 2.985 and 2.986. Since the new difference in absolute value is more than three times the previous difference, a 3-decimal place approximate solution to our system of equations is the value obtained in the previous calculation:

$$
x(2.985) \quad y(2.272 \text { value pair }
$$[^0]

Note. In practice, we would rather calculate than draw an enlarged diagram: from $x=2.95$ to $x=3.00$, i.e., increasing by 0.05, the surplus decreases from 0.044 to $(-0.018)$, i.e., by 0.062. Therefore, to make the decrease exactly 0.044, we need to increase $x$ by

$$
\frac{0.044}{0.062} \cdot 0.05=0.0355
$$

to $x=2.9855$ (assuming a linear change) and so on.


[^0]:    ${ }^{1}$ Some readers may see the $f(x)$ function notation for the first time here; therefore, we recommend that they get used to reading it as "function $f$ of $x$" rather than the commonly exaggerated "f of x". Here, $f$ and then $g$ indicate different dependencies on $x$. (Consider that we would also read $f x$ as "f of x"; those who avoid potential misunderstandings would say "f indexed by x" here.) The correct and unambiguous reading of $f(3)$ is "f at 3", or better: "f at $x=3$". - Similarly: sin $x$ : sine of $x$. - Eventually, of course, "f of x" is sufficient. Ed.