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Problem 637

# What is the slope of the line that passes through the origin and point $(1,2) ?$ the slope of a horizontal line? a vertical line?

Expert verified
The slope of the line passing through the origin and the point (1, 2) is 2. The slope of a horizontal line is 0, and the slope of a vertical line is undefined.
See the step by step solution

## Step 1: Identify the points

The two points we have are: - Origin: $$(0, 0)$$ - The given point: $$(1, 2)$$

## Step 2: Apply the slope formula

To find the slope of the line passing through these two points, we apply the slope formula, which is: $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ In this case: - $$x_{1} = 0$$, $$y_{1} = 0$$ - $$x_{2} = 1$$, $$y_{2} = 2$$ Plug these values into the formula: $m = \frac{2 - 0}{1 - 0}$

## Step 3: Calculate the slope

After plugging in the values, simplify the expression to find the slope: $m = \frac{2}{1}$ The slope of the line passing through the origin and (1, 2) is 2.

## Step 4: Discuss horizontal and vertical line slopes

For a horizontal line, the y-coordinates are the same, which means the change in y-coordinates is always zero. Therefore, the slope of a horizontal line is: $m = \frac{0}{(x_{2} - x_{1})} = 0$ For a vertical line, the x-coordinates are always the same, which means the change in x-coordinates is always zero. In this case, the slope is undefined, as dividing by zero results in an undefined value. So the slope of a vertical line is undefined. To summarize, the slope of the line passing through the origin and the point (1, 2) is 2, the slope of a horizontal line is 0, and the slope of a vertical line is undefined.

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