Math  /  Algebra

QuestionLesson 2 Checkpoint EVALUATE Independent Practice \begin{tabular}{|l|l|l|} \hline Learning Goal & Lesson Reflection (circle one) \\ \hline \begin{tabular}{l} I can describe and interpret the solution set of a \\ system of equations graphically and relate that to \\ the algebraic solution. \end{tabular} & Starting... Getting There... Got it! \\ \hline \end{tabular} \square \quad Complete the previous problems, check your solutions, then complete the Lesson Checkpoint below. Complete the Lesson Reflection above by circling your current understanding of the Learning Goal(s).
Solve f(x)=g(x)f(x)=g(x) by any method. Plot the point that represents the solution to the equation f(x)=g(x)f(x)=g(x).
1. Big City High School needs a baseball coach. Coach A is offering his services for an initial $5,000\$ 5,000 in addition to $450\$ 450 per session. Coach B is offering her services for an initial $4,000\$ 4,000 in addition to $700\$ 700 per session. When will the two coaches charge the same amount, and how much will it cost?

Interpret the solution in context.
2. Consider the following two equations: f(x)=4x6g(x)=12x+3\begin{array}{l} f(x)=-4 x-6 \\ g(x)=\frac{1}{2} x+3 \end{array}

Plot the point that represents the solution to the equation f(x)=g(x)f(x)=g(x). lifelong Algebra 1A (2024) Module 3 26

Studdy Solution

STEP 1

What is this asking? When will two coaches charge the same amount of money for their services, and how much will it cost?
Also, where do two lines intersect on a graph? Watch out! Don't mix up the coaches' fees!
Make sure you're using the right numbers for the initial fees and the per-session costs.
Also, remember that solving for xx gives you the number of sessions, not the total cost!

STEP 2

1. Set up the equations
2. Find the number of sessions
3. Calculate the total cost
4. Solve the second system of equations
5. Plot the point of intersection

STEP 3

Let's **define** what our variables represent.
We'll use f(x)f(x) to represent the **total cost** for Coach A, g(x)g(x) for the **total cost** for Coach B, and xx for the **number of sessions**.

STEP 4

Now we can **write** our equations.
Coach A charges an **initial $5,000\$5,000 plus $450\$450 per session**, so f(x)=450x+5000f(x) = 450x + 5000.
Coach B charges an **initial $4,000\$4,000 plus $700\$700 per session**, so g(x)=700x+4000g(x) = 700x + 4000.

STEP 5

To find when the coaches charge the same amount, we **set** f(x)f(x) **equal** to g(x)g(x): 450x+5000=700x+4000450x + 5000 = 700x + 4000.

STEP 6

Now, let's **solve for** xx.
We can **subtract** 450x450x from both sides to get 5000=250x+40005000 = 250x + 4000.
Then, **subtract** 40004000 from both sides to get 1000=250x1000 = 250x.
Finally, **divide** both sides by 250250 to find x=4x = 4.
This means the coaches will charge the same amount after **4 sessions**.

STEP 7

Now that we know the **number of sessions**, we can **plug** x=4x = 4 back into either equation to find the **total cost**.
Let's use f(x)f(x): f(4)=4504+5000=1800+5000=6800f(4) = 450 \cdot 4 + 5000 = 1800 + 5000 = 6800.
So, the **total cost** will be $6,800\$6,800.

STEP 8

We are given f(x)=4x6f(x) = -4x - 6 and g(x)=12x+3g(x) = \frac{1}{2}x + 3.
To find where these lines intersect, we **set** f(x)=g(x)f(x) = g(x), so 4x6=12x+3-4x - 6 = \frac{1}{2}x + 3.

STEP 9

Let's **solve for** xx. **Add** 4x4x to both sides: 6=92x+3-6 = \frac{9}{2}x + 3.
Then, **subtract** 33 from both sides: 9=92x-9 = \frac{9}{2}x. **Multiply** both sides by 29\frac{2}{9} to get x=2x = -2.

STEP 10

Now, **substitute** x=2x = -2 into either equation to find the corresponding yy value.
Let's use g(x)g(x): g(2)=12(2)+3=1+3=2g(-2) = \frac{1}{2}(-2) + 3 = -1 + 3 = 2.

STEP 11

The solution to the second system is the point (2,2)(-2, 2).
This is where the two lines intersect on a graph.

STEP 12

The two coaches will charge the same amount after **4 sessions**, and the total cost will be $6,800\$6,800.
The solution to the second system of equations is the point (2,2)(-2, 2).

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