Write the equation of the line, in standard form, with the given conditions: The line passes through (1,2) and intersects the x – and y – axes at (c, 0) and (0, c), respectively.

To write the equation of the line that passes through the point \((1, 2)\) and intersects the x-axis and y-axis at \((c, 0)\) and \((0, c)\), respectively, we need to follow these steps:

### Step 1: Determine the slope of the line
The slope \(m\) of the line that passes through the points \((c, 0)\) and \((0, c)\) can be calculated using the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{c - 0}{0 - c} = -1
\]
So, the slope \(m\) is \(-1\).

### Step 2: Use the point-slope form of the equation of a line
The point-slope form of the equation of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
We can use the point \((1, 2)\) and the slope \(m = -1\):
\[
y - 2 = -1(x - 1)
\]

### Step 3: Simplify to find the equation in slope-intercept form
Expand and simplify the equation:
\[
y - 2 = -x + 1
\]
\[
y = -x + 3
\]

### Step 4: Convert the equation to standard form
The standard form of a line equation is \(Ax + By = C\). Rearrange the equation \(y = -x + 3\) to this form:
\[
x + y = 3
\]

So, the equation of the line in standard form is:
\[
\boxed{x + y = 3}
\]

This equation confirms that the line passes through the point \((1, 2)\) and intersects the axes at \((3, 0)\) and \((0, 3)\).

Would you like more details on this problem, or do you have any questions? Here are some related questions you might be interested in:

1. How can you verify if a point lies on a given line?
2. How do you find the x- and y-intercepts of a linear equation?
3. What is the general method to convert slope-intercept form to standard form?
4. Can a line have a slope of zero? What would that look like?
5. What are the properties of parallel and perpendicular lines in terms of their slopes?

**Tip:** When converting from slope-intercept to standard form, always ensure the coefficients \(A\), \(B\), and \(C\) are integers, and \(A\) is non-negative.

Write the equation of the line, in standard form, with the given conditions: The line passes through (1,2) and intersects the x- and y-axes at (c, 0) and (0, c), respectively.

Learn how to write the equation of a line that passes through the point (1,2) and intersects the x- and y-axes in standard form. This problem covers key concepts in algebra and coordinate geometry, making it suitable for students in Grades 9-10.

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