Which is an equation of the line through $(-1,-4)$ and parallel to the line $3 x+y=5$?
The goal is to find the equation of a line that passes through the point \((-1, -4)\) and is parallel to the given line \(3x + y = 5\). We know that parallel lines have the same slope. Therefore, we first need to identify the slope of the given line. We can do this by converting the equation \(3x + y = 5\) into the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope. Subtracting \(3x\) from both sides gives \(y = -3x + 5\). Thus, the slope of the line is \(-3\).
Now that we have the slope \(m = -3\) and the point \((-1, -4)\) that the new line must pass through, we can use the point-slope form of a linear equation, which is given by \(y - y_0 = m(x - x_0)\), where \((x_0, y_0)\) is the given point. Substituting our values, we get \(y - (-4) = -3(x - (-1))\). This simplifies to \(y + 4 = -3(x + 1)\).
Finally, we simplify the equation from step 2 to express it in slope-intercept form. Distributing the \(-3\) on the right side gives \(y + 4 = -3x - 3\). Subtracting 4 from both sides results in \(y = -3x - 7\). Comparing this result to the given options, we find it matches option c.
In real life, engineers and architects use the property of parallel lines to design structures. For example, when building a staircase, the handrail must be parallel to the path of the stairs to ensure safety and comfort. By knowing the slope of the staircase, one can use the point-slope form to calculate the precise equation for the handrail's path, ensuring it is perfectly parallel and correctly positioned.