In the xy-plane, two lines are graphed as shown. The lines represent a system of linear equations whose solution is...

1. INFER the key concept

When you have a system of two linear equations graphed on the same coordinate plane, the solution is the point where the two lines intersect. This is because the intersection point is the only ordered pair \(\mathrm{(x, y)}\) that satisfies both equations simultaneously.

Key insight: You don't need to find equations or do algebra—just locate where the lines cross!

2. TRANSLATE the visual information

Looking at the graph:

• Two lines are shown: \(\mathrm{L1}\) and \(\mathrm{L2}\)
• \(\mathrm{L1}\) appears to be V-shaped (with a vertex)
• \(\mathrm{L2}\) is a straight line with positive slope

Find the intersection: Trace each line and identify where they meet. Looking carefully at the grid:

• The lines intersect at \(\mathrm{x = -1}\) (one unit to the left of the origin)
• At this x-value, both lines pass through \(\mathrm{y = 3}\) (three units above the origin)
• The intersection point is \(\mathrm{(-1, 3)}\)

3. TRANSLATE to answer the specific question

The question asks: "What is the value of y?"

From the intersection point \(\mathrm{(-1, 3)}\):

• \(\mathrm{x = -1}\)
• \(\mathrm{y = 3}\)

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread the graph and confuse the x-coordinate with the y-coordinate.

The intersection point is \(\mathrm{(-1, 3)}\), but a student might see that the x-value is -1 and incorrectly answer -1 instead of reading the y-coordinate, which is 3. This happens because students rush through reading "what is the value of y?" and don't carefully identify which coordinate the question is asking for.

This leads to entering -1 as the answer instead of 3.

Second Most Common Error:

Weak TRANSLATE skill: Students miscount grid lines or misjudge the intersection point.

If a student isn't careful when reading the y-coordinate, they might think the intersection occurs at \(\mathrm{y = 2}\) or \(\mathrm{y = 4}\) instead of \(\mathrm{y = 3}\). This can happen if they:

• Start counting from 1 instead of 0
• Misalign their visual reading with the grid lines
• Estimate the position incorrectly

This leads to entering 2 or 4 as the answer.

The Bottom Line:

This problem tests whether students understand the fundamental graphical interpretation of systems of equations (that solutions occur at intersection points) and can accurately read coordinates from a graph. The key is to slow down and carefully identify: (1) where the lines cross, and (2) which coordinate value the question is asking for.