Let's break down the two questions shown in the image.

Question 7: System of Linear Equations

• Siblings' savings:
  • Sibling A's savings: $y = 10x + 25$
  • Sibling B's savings: $y = 5x + 50$

We are asked to determine when their savings will be the same (i.e., when the two equations intersect). To find this, set the equations equal to each other:

$10x + 25 = 5x + 50$

Now, solve for $x$:

$10x - 5x = 50 - 25$

$5x = 25$

$x = 5$

Thus, after 5 weeks, both siblings will have the same amount of money.

Correct answer: 5 weeks

Question 8: Rotation of Coordinates

The triangle has vertices at:

• $U(-1, 1)$
• $V(0, -4)$
• $W(-4, -1)$

We need to rotate the triangle 90° counterclockwise and determine the new coordinates.

A 90° counterclockwise rotation rule is:

• $(x, y)$ becomes $(-y, x)$

Apply this rotation to each vertex:

• $U(-1, 1)$ becomes $( -1, 1 ) \to (-1, -1)$
• $V(0, -4)$ becomes $( 0, -4 ) \to (4, 0)$
• $W(-4, -1)$ becomes $( -4, -1 ) \to (1, -4)$

So the new coordinates are:

• $U'(-1, -1)$
• $V'(4, 0)$
• $W'(1, -4)$

Correct answer: Option 2: $U'(-1, -1), V'(4, 0), W'(1, -4)$

Would you like more details on any of the steps? Let me know if you have any further questions!

Related Questions:

1. How do you solve for the intersection point of two linear equations?
2. What is the general method for performing a rotation of coordinates?
3. How do different rotations (90°, 180°, 270°, etc.) affect coordinates?
4. How do systems of equations help model real-life problems?
5. What happens if the slope or y-intercept of one of the equations changes?

Tip: When solving systems of equations, always isolate the variable first (if possible), and check your solution by substituting it back into both original equations to ensure they are equal.