Question

question
a law firm is going to designate associates and partners to a big new case. the daily rate charged to the client for each associate is $400 and the daily rate for each partner is $1200. the law firm assigned a total of 5 lawyers to the case and was able to charge the client $3600 per day for these lawyers’ services. graphically solve a system of equations in order to determine the number of associates assigned to the case, x, and the number partners assigned to the case, y.
click twice to plot each line. click a line to delete it.

Explanation:

Step1: Define Variables and Equations

Let \( x \) be the number of associates and \( y \) be the number of partners.
From the total number of lawyers: \( x + y = 5 \).
From the total daily charge: \( 400x + 1200y = 3600 \). Simplify the second equation by dividing by 400: \( x + 3y = 9 \).

Step2: Graph the First Equation (\( x + y = 5 \))

Rewrite in slope - intercept form (\( y=mx + b \)): \( y=-x + 5 \).

• When \( x = 0 \), \( y = 5 \) (plot the point \((0,5)\)).
• When \( y = 0 \), \( x = 5 \) (plot the point \((5,0)\)).

Draw a line through these two points.

Step3: Graph the Second Equation (\( x + 3y = 9 \))

Rewrite in slope - intercept form: \( 3y=-x + 9\Rightarrow y=-\frac{1}{3}x + 3 \).

• When \( x = 0 \), \( y = 3 \) (plot the point \((0,3)\)).
• When \( y = 0 \), \( x = 9 \) (but since \( x + y=5 \) and \( x,y\geq0 \), we can also use another point. Let \( x = 3 \), then \( 3+3y = 9\Rightarrow y = 2 \), so plot the point \((3,2)\)).

Draw a line through these points.

Step4: Find the Intersection

The two lines intersect where the equations are equal. Solve the system:
\(

$$\begin{cases}y=-x + 5\\y=-\frac{1}{3}x + 3\end{cases}$$

\)
Set \(-x + 5=-\frac{1}{3}x + 3\).
\(-x+\frac{1}{3}x=3 - 5\).
\(-\frac{2}{3}x=-2\).
\(x = 3\).
Substitute \( x = 3 \) into \( y=-x + 5 \), we get \( y=2 \).

Answer:

The number of associates \( x = 3 \) and the number of partners \( y = 2 \). So the solution to the system is \( x = 3 \), \( y = 2 \).