Answer :
To solve this problem, we need to find a system of equations that accurately describes the situation given in the question. Let's break it down:
1. Understanding the components:
- We have a total of 25 problems.
- Some problems are worth 2 points each.
- Some problems are worth 3 points each.
- The total number of points for the test is 60.
2. Defining the variables:
- Let [tex]\( x \)[/tex] represent the number of 2-point problems.
- Let [tex]\( y \)[/tex] represent the number of 3-point problems.
3. Setting up the equations:
- The total number of problems is 25, so:
[tex]\[
x + y = 25
\][/tex]
- The total number of points is 60, so:
[tex]\[
2x + 3y = 60
\][/tex]
Now, let's look at the options provided to find the system that matches these equations:
- Option (A):
[tex]\[
\left\{
\begin{array}{l}
x + y = 25 \\
2x + 3y = 60
\end{array}
\right.
\][/tex]
This system matches our equations perfectly.
- Option (B):
[tex]\[
\left\{
\begin{array}{l}
x + y = 60 \\
2x + 3y = 25
\end{array}
\right.
\][/tex]
This option swaps the roles of total points and total problems, which does not match our situation.
- Option (C):
[tex]\[
\left\{
\begin{array}{l}
x - y = 25 \\
2x + 3y = 60
\end{array}
\right.
\][/tex]
The first equation in this set does not reflect the total number of problems correctly.
- Option (D):
[tex]\[
\left\{
\begin{array}{l}
x - y = 60 \\
2x - 3y = 25
\end{array}
\right.
\][/tex]
Both equations in this option do not represent the given conditions correctly.
The correct system is Option (A), which uses the equations:
[tex]\[
x + y = 25
\][/tex]
[tex]\[
2x + 3y = 60
\][/tex]
This system will accurately help determine the number of 2-point problems and 3-point problems on the test.
1. Understanding the components:
- We have a total of 25 problems.
- Some problems are worth 2 points each.
- Some problems are worth 3 points each.
- The total number of points for the test is 60.
2. Defining the variables:
- Let [tex]\( x \)[/tex] represent the number of 2-point problems.
- Let [tex]\( y \)[/tex] represent the number of 3-point problems.
3. Setting up the equations:
- The total number of problems is 25, so:
[tex]\[
x + y = 25
\][/tex]
- The total number of points is 60, so:
[tex]\[
2x + 3y = 60
\][/tex]
Now, let's look at the options provided to find the system that matches these equations:
- Option (A):
[tex]\[
\left\{
\begin{array}{l}
x + y = 25 \\
2x + 3y = 60
\end{array}
\right.
\][/tex]
This system matches our equations perfectly.
- Option (B):
[tex]\[
\left\{
\begin{array}{l}
x + y = 60 \\
2x + 3y = 25
\end{array}
\right.
\][/tex]
This option swaps the roles of total points and total problems, which does not match our situation.
- Option (C):
[tex]\[
\left\{
\begin{array}{l}
x - y = 25 \\
2x + 3y = 60
\end{array}
\right.
\][/tex]
The first equation in this set does not reflect the total number of problems correctly.
- Option (D):
[tex]\[
\left\{
\begin{array}{l}
x - y = 60 \\
2x - 3y = 25
\end{array}
\right.
\][/tex]
Both equations in this option do not represent the given conditions correctly.
The correct system is Option (A), which uses the equations:
[tex]\[
x + y = 25
\][/tex]
[tex]\[
2x + 3y = 60
\][/tex]
This system will accurately help determine the number of 2-point problems and 3-point problems on the test.
