Question

Write an equation in standard form of the line that passes through the given point and has the given slope. $$(-4,2), m=-2$$

Short answer

The equation in standard form of the line that passes through the point \((-4,2)\), with slope \(-2\), is \(2x + y = -6\).

Step-by-step solution

Substitute the given values into the point-slope formula

First, plug the values \((-4,2)\) into the point-slope formula for \(x_{1}, y_{1}\) and \(m=-2\) for the slope which will give us \(y - 2 = -2(x - (-4))\).

Simplify the equation

Simplify the equation to derive the slope-intercept form, \(y = mx + c\). The calculation simplifies to \(y = -2x - 6\).

Convert to Standard Form

The equation in slope-intercept form is \(y = -2x -6\). Now, we convert this into standard form. This involves removing any fractions, making 'A' positive and ensuring that A, B, and C are integers. Rearranging and simplifying the equation gives the standard form as \(2x + y = -6\).

Key concepts

Point-Slope Formula

Understanding the point-slope formula is essential for writing equations of lines when you know a point on the line and its slope. The formula is expressed as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope of the line, and \(x_1\), \(y_1\) are the coordinates of the known point. This compact equation forms a bridge to other forms of linear equations.

For instance, if we have a point \( (-4,2) \) and a slope \(m=-2\), the point-slope form based on this information would be \(y - 2 = -2(x + 4)\). This formula shows how the slope of the line influences the rise and run between any point on the line and the known point. From there, you can expand and simplify to get the equation in other forms. It’s essentially the first step in the linear equation journey!

Slope-Intercept Form

The slope-intercept form, represented as \(y = mx + b\), is one of the most commonly used representations of a linear equation. This form clearly indicates the slope of the line (\(m\)) and the y-intercept (\(b\)), the point where the line crosses the y-axis.

Using our example, after applying the point-slope formula, we can simplify to get the slope-intercept form \(y = -2x - 6\). This shows that for every unit increase in \(x\), the value of \(y\) decreases by 2, since the slope is \(m=-2\). The y-intercept here is \(b=-6\), which means the line crosses the y-axis at \(y=-6\). It's the equation you'll often use to graph a line quickly or to understand a line's trend.

Standard Form of a Linear Equation

Finally, we have the standard form of a linear equation, which is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) is non-negative. When converting to standard form, it's important to rearrange the equation so that \(x\) and \(y\) are on the same side and \(x\) is positive.

Let's take our slope-intercept equation \(y = -2x - 6\) and convert it to standard form by rearranging the terms to get \(2x + y = -6\). This form is particularly useful for analyzing the intercepts and is preferred for certain algebraic operations. Here, the coefficients \(A=2\) and \(B=1\), with \(C=-6\), all of which are integers, and \(A\) is positive as required.