Question

1. *

which graph represents the system of equations? ( y = \frac{2}{3}x + 1 )
( y = -\frac{1}{3}x + 1 )

four graphs labeled a, b, c, d with corresponding multiple - choice options below them:
( \bigcirc ) a
( \bigcirc ) b
( \bigcirc ) c
( \bigcirc ) d

Explanation:

Step1: Analyze the y - intercepts

For a linear equation in the form \(y = mx + b\), \(b\) is the y - intercept. Both equations \(y=\frac{2}{5}x + 1\) and \(y=-\frac{1}{5}x + 1\) have a y - intercept of \(b = 1\). This means both lines should cross the y - axis at the point \((0,1)\).

Step2: Analyze the slopes

The slope \(m\) of a line \(y=mx + b\) determines the direction and steepness of the line. For \(y=\frac{2}{5}x+1\), the slope \(m=\frac{2}{5}\) (positive, so the line rises from left to right). For \(y =-\frac{1}{5}x + 1\), the slope \(m=-\frac{1}{5}\) (negative, so the line falls from left to right).

Now let's analyze the options:

• Option A: The lines do not seem to have the correct slopes or y - intercepts. The slopes appear too flat or not matching the given slopes.
• Option B: One line has a very steep slope, which does not match the slopes \(\frac{2}{5}\) and \(-\frac{1}{5}\).
• Option C: Both lines cross the y - axis at \((0,1)\) (same y - intercept). One line has a positive slope (rising from left to right) and the other has a negative slope (falling from left to right), which matches the slopes \(\frac{2}{5}\) and \(-\frac{1}{5}\).
• Option D: The lines do not cross the y - axis at the same point, so this is incorrect.

Answer:

C