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The point-slope form is a way to write the equation of a line. It's particularly useful when you know a point on the line and the slope, but not necessarily the intercepts. The formula is given by: \[ y - y_1 = m(x - x_1) \] where

• \( (x_1, y_1) \) is a point on the line
• m represents the slope of the line.

By using this form, you can quickly determine the equation of a line just by plugging in known values for a given point and slope.
For instance, in the given problem, we used the point \((6, -1)\) and the slope \(m\) to form the equation \(y + 1 = m(x - 6)\). This equation can then be expanded and rearranged to match the standard form \(y = mx + c\). By using the point-slope form, you're employing a flexible approach that works well under various scenarios.

A quadratic equation is a polynomial equation of degree two. Its standard form looks like this:\[ ax^2 + bx + c = 0 \] where

• \(a, b,\) and \(c\) are constants.
• \(a eq 0\).

A quadratic equation is solved to find the values of \(x\) that satisfy this equation, known as the roots of the equation.
In the solution provided, the expression \(36m^2 + 15m + 1 = 0\) is an example of a quadratic equation in terms of \(m\). Solving it yields the factorization as \((12m + 1)(3m + 1) = 0\) which can be broken down to find that \(m = -\frac{1}{12}\) or \(m = -\frac{1}{3}\). These roots help us determine specific characteristics of the lines in the original problem.

The y-intercept is the point where the line crosses the y-axis on a graph. For the equation of a line \(y = mx + c\), the y-intercept is the value of \(c\).
This is because, at the y-intercept, \(x\) is zero, making \(y = c\).
In our problem, we found different values of \(c\) depending on the slope \(m\). By solving the quadratic equation \((-6m - 1)^2 = -3m\), the value \(c\) was determined via simplifying to find \(c = -6m - 1\). The values of \(m\) we discovered tell us the y-intercepts:

• For \(m = -\frac{1}{12}\), the y-intercept \(c\) is \(-\frac{1}{2}\).
• For \(m = -\frac{1}{3}\), we found \(c = 1\).

Understanding the y-intercept helps us see where the line crosses the y-axis, providing a starting point for sketching the line on a graph.

The x-intercept of a line is the point at which the line crosses the x-axis. At the x-intercept, the value of \(y\) is zero.
For an equation given as \(y = mx + c\), setting \(y\) to zero gives \(mx + c = 0\). This can be solved for \(x\), resulting in \[ x = -\frac{c}{m} \].
In the provided example, the product of the x- and y-intercepts was specified to be 3, leading to the condition \(-\frac{c}{m} \cdot c = 3\) or \(c^2 = -3m\). This condition is key to solving the overall problem.
By determining \(c\) and \(m\), we calculated the x-intercepts for each equation:

• For \(m = -\frac{1}{12}\), we can substitute \(c = -\frac{1}{2}\) into \(x = -\frac{c}{m}\) to find the x-intercept.
• Similarly, for \(m = -\frac{1}{3}\), using \(c = 1\), we found another x-intercept.

Knowing the x-intercept allows us to understand where the line crosses the x-axis, further aiding in graphing the equation.