Question

1. lines a and b are shown on the grid.

the equation of line a is \\(y = 5x + 40\\)
find the equation of line b

Explanation:

Determine grid scale

The equation of line A is given as:

$$y = 5x + 40$$

From the graph, line A crosses the y-axis at a grid intersection.
According to the equation, the y-intercept is \(40\).
Counting grid units from the origin \(O(0,0)\) up to the y-intercept of line A:
There are exactly \(8\) grid squares.
Thus, each grid unit along the y-axis represents:

$$\frac{40}{8} = 5\text{ units}$$

Looking at the x-intercept of line A:
Setting \(y = 0\) in \(y = 5x + 40\) gives:

$$0 = 5x + 40 \implies x = -8$$

Counting grid units from the origin to the left along the x-axis to the x-intercept of line A:
There are exactly \(8\) grid squares.
Thus, each grid unit along the x-axis also represents:

$$\frac{8}{8} = 1\text{ unit}$$

So, each grid square is \(1\text{ unit}\) horizontally and \(5\text{ units}\) vertically.

Find y-intercept of line B

Line B crosses the y-axis below the origin.
Counting grid squares down from the origin to the y-intercept of line B:
It is exactly \(6\) grid squares down.
Since each vertical grid unit represents \(5\text{ units}\):

$$y\text{-intercept } (b) = -6 \times 5 = -30$$

Find another point on line B

Let's identify a clear grid intersection point on line B.
Looking at the x-axis, line B crosses the x-axis to the left of the origin.
Counting grid squares to the left from the origin to the x-intercept of line B:
It is exactly \(4\) grid squares to the left.
Since each horizontal grid unit represents \(1\text{ unit}\):

$$x\text{-intercept} = -4 \times 1 = -4$$

This gives the point \((-4, 0)\) on line B.

Calculate slope of line B

Using the two points on line B, \((0, -30)\) and \((-4, 0)\):

$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-30 - 0}{0 - (-4)} = \frac{-30}{4} = -7.5$$

Or in fractional form:

$$m = -\frac{15}{2}$$

Write equation of line B

Using the slope-intercept form \(y = mx + b\):

$$y = -7.5x - 30$$

Or in fractional form:

$$y = -\frac{15}{2}x - 30$$

Answer:

\(y = -7.5x - 30\)