Question

what does \solution to a system of equations\ mean?
a) it’s the point where graphs of both equations cross the y - axis
b) it’s the point that solves both equations at the same time
c) it’s the point where graphs of both equations cross the x - axis
d) it’s the point where both equations equal zero

options: a, b, c, d (radio buttons)

question 2
1 pts
solve the following systems of equations using substitution:
x = 6
y = 2x - 3

options: (9,9), (9,6), (6,9), (6,6) (radio buttons)

Explanation:

Question 1 (Multiple Choice)

• Option a: A system's solution isn't about crossing the y - axis. The y - axis crossing is the y - intercept of a single equation, not a system's solution.
• Option b: By definition, a solution to a system of equations is a set of values (a point in 2D) that satisfies all equations in the system simultaneously. So this option is correct.
• Option c: Crossing the x - axis is the x - intercept of a single equation, not related to a system's solution.
• Option d: A point where both equations equal zero would be a special case (like a common root), but the general definition of a system's solution is a point that satisfies all equations, not necessarily where they equal zero.

Step1: Substitute \( x = 6 \) into \( y=2x - 3 \)

We know that \( x = 6 \), so we replace \( x \) in the equation \( y = 2x-3 \) with 6. The equation becomes \( y=2\times6 - 3 \).

Step2: Calculate the value of \( y \)

First, calculate \( 2\times6=12 \), then \( y = 12-3=9 \). So when \( x = 6 \), \( y = 9 \), and the solution to the system is the point \( (x,y)=(6,9) \).

Answer:

b. It's the point that solves both equations at the same time

Question 2 (Solving System by Substitution)