a. In a card game, each round you earn either 3 points or 5 points depending on
the cards you play. After 5 rounds you have 19 points.
Use for the number of 3 point rounds and for the number of 5 point
rounds. Write a system of 2 equations that describes this situation.
b. Another system is solved by the point . Explain how you can check that
this solution is correct.

Question

a. In a card game, each round you earn either 3 points or 5 points depending on
the cards you play. After 5 rounds you have 19 points.
Use

for the number of 3 point rounds and

for the number of 5 point
rounds. Write a system of 2 equations that describes this situation.
b. Another system is solved by the point

. Explain how you can check that
this solution is correct.

UpStudy AI · Accepted Answer

### a. Writing the System of Equations

Let's define the variables:
- $$ x $$ = number of 3-point rounds
- $$ y $$ = number of 5-point rounds

Given the information:
1. **Total Rounds:** You played a total of 5 rounds.
2. **Total Points:** You earned a total of 19 points.

Using these, we can set up the following two equations:

1. **Total Rounds Equation:**
   $$
   x + y = 5
   $$
   *This equation states that the sum of 3-point rounds and 5-point rounds is equal to 5.*

2. **Total Points Equation:**
   $$
   3x + 5y = 19
   $$
   *This equation represents the total points earned, where each 3-point round contributes 3 points and each 5-point round contributes 5 points.*

**System of Equations:**
$$
\begin{cases}
x + y = 5 \
3x + 5y = 19
\end{cases}
$$

---

### b. Verifying the Solution $$(7, 10)$$

To check whether $$(7, 10)$$ is a valid solution to the system of equations, follow these steps:

1. **Substitute $$ x = 7 $$ and $$ y = 10 $$ into the First Equation:**
   $$
   7 + 10 = 5 \
   17 = 5 \quad \text{(This is FALSE)}
   $$
   
2. **Substitute $$ x = 7 $$ and $$ y = 10 $$ into the Second Equation:**
   $$
   3(7) + 5(10) = 19 \
   21 + 50 = 19 \
   71 = 19 \quad \text{(This is FALSE)}
   $$

Since substituting $$(7, 10)$$ into both equations does not satisfy either equation, $$(7, 10)$$ is **not** a solution to the system.

**Note:** It's possible that $$(7, 10)$$ refers to a different system of equations. To verify it correctly:
- **Identify the Specific System:** Ensure you know which equations you're testing.
- **Substitute the Values:** Plug $$ x = 7 $$ and $$ y = 10 $$ into each equation of that system.
- **Check for Validity:** Both equations should hold true after substitution.

If both equations are satisfied after substitution, then $$(7, 10)$$ is a valid solution. Otherwise, it is not.
a. The system of equations is: $$ \begin{cases} x + y = 5 \ 3x + 5y = 19 \end{cases} $$ b. To check if $$(7, 10)$$ is a solution: - Substitute $$ x = 7 $$ and $$ y = 10 $$ into the first equation: $$ 7 + 10 = 17 \neq 5 $$ → Not valid. - Substitute into the second equation: $$ 3(7) + 5(10) = 21 + 50 = 71 \neq 19 $$ → Not valid. So, $$(7, 10)$$ is **not** a solution.

Upstudy AI Solution

Answer

Solution

a. Writing the System of Equations

b. Verifying the Solution

Extra Insights

Related Questions

Latest Algebra Questions