Question

6 systems of equations: word problem practice objective: use the desmos graphing calculator to solve these real - world scenarios. for the first five, the equations are provided. for the last five, you must write the equations before graphing. the chemistry mix a scientist is mixing two saline solutions to get a specific concentration. the total volume of the mix is 12 liters ($x + y = 12$). the first solution is 20% salt and the second is 50% salt, and she needs the final mix to have 3.6 liters of salt ($0.20x + 0.50y = 3.6$). equations: $x + y = 12$ and $0.2x + 0.5y = 3.6$

Explanation:

Step1: Solve the first equation for \( x \)

From \( x + y = 12 \), we can express \( x \) as \( x = 12 - y \).

Step2: Substitute \( x = 12 - y \) into the second equation

Substitute into \( 0.2x + 0.5y = 3.6 \):
\[

$$\begin{align*} 0.2(12 - y) + 0.5y &= 3.6\\ 2.4 - 0.2y + 0.5y &= 3.6\\ 2.4 + 0.3y &= 3.6 \end{align*}$$

\]

Step3: Solve for \( y \)

Subtract 2.4 from both sides:
\[
0.3y = 3.6 - 2.4 = 1.2
\]
Divide both sides by 0.3:
\[
y = \frac{1.2}{0.3} = 4
\]

Step4: Solve for \( x \)

Substitute \( y = 4 \) into \( x = 12 - y \):
\[
x = 12 - 4 = 8
\]

Answer:

\( x = 8 \) liters of the 20% salt solution and \( y = 4 \) liters of the 50% salt solution.