Question

a graph and a table view for a system of equations are shown.
what do you notice?
i notice_____________________________________________________
_____________________________________________________
_____________________________________________________.
what do you wonder?
i wonder _____________________________________________________
_____________________________________________________
_____________________________________________________.
vocabulary
a system of linear equations is a set of two or more linear equations containing two or more variables
examples: $y = 3x - 1$ and $y = 2x - 1$
the solution of systems of linear equations is the ordered pair $(x, y)$ that satisfies both equations in the system

Explanation:

For "What do you notice?"

From the graph, the two lines \( Y_1 = X - 2 \) and \( Y_2=-2X + 7 \) intersect at a point. From the table, as \( X \) increases, \( Y_1 \) increases (since its slope is positive) and \( Y_2 \) decreases (since its slope is negative). Also, we can look for when \( Y_1 = Y_2 \) (the solution of the system). Let's solve \( X - 2=-2X + 7 \). Adding \( 2X \) to both sides: \( 3X-2 = 7 \), then adding 2: \( 3X=9 \), so \( X = 3 \). Then \( Y_1=3 - 2=1 \), \( Y_2=-2(3)+7 = 1 \). So at \( X = 3 \), \( Y_1 = Y_2 = 1 \), meaning the lines intersect at \( (3,1) \). Also, the slope of \( Y_1 \) is 1 (positive, increasing) and slope of \( Y_2 \) is -2 (negative, decreasing). The y - intercept of \( Y_1 \) is -2 (when \( X = 0 \), \( Y=-2 \)) and y - intercept of \( Y_2 \) is 7 (when \( X = 0 \), \( Y = 7 \)).

Possible wonders could be about other points of intersection (but since they are linear, there's only one), how the slopes affect the intersection, what happens if we change the equations (like changing the slope or y - intercept), or how to find the solution algebraically (though we did that, but a student might wonder). Also, wonder about the meaning of the solution in a real - world context if these equations represented real - world quantities.

Answer:

I notice that the two lines \( Y_1 = X - 2 \) and \( Y_2=-2X + 7 \) intersect at the point \( (3,1) \) (from solving or checking the table/graph), \( Y_1 \) has a positive slope (increasing) and \( Y_2 \) has a negative slope (decreasing), and at \( X = 0 \), \( Y_1=-2 \) and \( Y_2 = 7 \).

For "What do you wonder?"