Let's analyze and solve the given system of equations step-by-step:
The system of equations is:
1. [tex]\( I + k = 900 \)[/tex]
2. [tex]\( I = k + 250 \)[/tex]
Here, [tex]\( I \)[/tex] represents the points that Lia scored, and [tex]\( k \)[/tex] represents the points that Kate scored.
Step 1: Express Lia's points in terms of Kate's points from the second equation:
From the second equation, we have:
[tex]\[ I = k + 250 \][/tex]
Step 2: Substitute this expression into the first equation:
Replace [tex]\( I \)[/tex] in the first equation [tex]\( I + k = 900 \)[/tex] with [tex]\( k + 250 \)[/tex]:
[tex]\[ k + 250 + k = 900 \][/tex]
Step 3: Simplify and solve for [tex]\( k \)[/tex]:
Combine like terms:
[tex]\[ 2k + 250 = 900 \][/tex]
Subtract 250 from both sides:
[tex]\[ 2k = 900 - 250 \][/tex]
[tex]\[ 2k = 650 \][/tex]
Divide by 2 to solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{650}{2} \][/tex]
[tex]\[ k = 325 \][/tex]
So, Kate scored 325 points.
Step 4: Solve for Lia's points using the second equation:
[tex]\[ I = k + 250 \][/tex]
Substitute [tex]\( k = 325 \)[/tex] into the equation:
[tex]\[ I = 325 + 250 \][/tex]
[tex]\[ I = 575 \][/tex]
So, Lia scored 575 points.
Conclusion:
- Kate scored 325 points.
- Lia scored 575 points.
Thus, the correct choice is:
Kate [tex]\( = 325 \)[/tex] and Lia [tex]\( = 575 \)[/tex].
