Question

which graph represents the system of equations?
$y = \frac{2}{5} x + 1$
$y = -\frac{1}{5} x + 1$
(there are four graphs labeled a, b, c, d with corresponding radio buttons for selection)

Explanation:

Step1: Analyze the y - intercepts

For a linear equation in the form \(y = mx + b\), the \(y\) - intercept is \(b\). For both equations \(y=\frac{2}{5}x + 1\) and \(y=-\frac{1}{5}x + 1\), the \(y\) - intercept \(b = 1\). This means both lines 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 the line \(y=\frac{2}{5}x + 1\), the slope \(m=\frac{2}{5}>0\). A positive slope means the line rises from left to right.
• For the line \(y =-\frac{1}{5}x+1\), the slope \(m =-\frac{1}{5}<0\). A negative slope means the line falls from left to right.

Now let's analyze the options:

• Option A: Both lines have a \(y\) - intercept of 1 (cross the \(y\) - axis at the same point), one line has a positive slope (rises from left to right) and the other has a negative slope (falls from left to right), which matches our analysis.
• Option B: The lines seem to cross at the origin \((0,0)\) instead of \((0,1)\), so it does not match.
• Option C: The lines do not seem to have the same \(y\) - intercept, so it does not match.
• Option D: The lines do not seem to have the same \(y\) - intercept, so it does not match.

Answer:

A