Question

Write an equation in standard form of the line that passes through the two points. $$(-4,1),(2,-5)$$

Short answer

The equation of the line passing through (-4,1) and (2,-5) in standard form is \(x + y = -3\).

Step-by-step solution

Calculate The Slope

The formula for calculating the slope (m) of a line between two given points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). We apply this formula to our points (-4,1) and (2,-5), and get \(m = \frac{(-5) - 1}{2 - (-4)} = -1\).

Find the Equation Using Point-Slope form

Next, we plug our slope and one of our points into the point-slope form of a line equation, which is \(y - y_1 = m(x - x_1)\). Using the point (2,-5) and the slope -1, we get \(y - (-5) = -1(x - 2)\). Simplify to get \(y + 5 = -x + 2\).

Convert to Standard Form

The standard form of a linear equation is \(Ax + By = C\). We can rearrange the equation we found to this form by adding x to both sides and subtracting 5 from both sides: \(y + x = -3\). So, the standard form of the line passing through (-4,1) and (2,-5) is \(x + y = -3\).

Key concepts

Calculating Slope

When it comes to understanding linear equations, the slope is a foundational concept. The slope measures the steepness of a line and shows how much the y-value of a point on the line changes as the x-value increases by one unit. In simple terms, it tells us how 'tilted' the line is.
The formula for calculating the slope, denoted as 'm', requires two distinct points on the line, \( (x_1, y_1) \) and \( (x_2, y_2) \). It's expressed mathematically as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
For example, given two points (-4, 1) and (2, -5), we can plug these coordinates into the slope formula to find \( m = \frac{(-5) - 1}{2 - (-4)} \), which simplifies down to \( m = -1 \). This negative slope indicates that the line falls as it moves from left to right. Understanding this concept is crucial as it's not only central to linear equations but also to other areas of algebra and calculus.

Point-Slope Form

Once we have the slope of a line, the next step is often writing its equation. One particularly useful form for this is the point-slope form, especially when we have a point through which the line passes and its slope.
The point-slope form of a linear equation is written as \( y - y_1 = m(x - x_1) \), where \(m\) is the slope and \( (x_1, y_1) \) is the point on the line. It's a direct way to use the given information to write an equation.
Let's take the point (2, -5) and the slope -1 from our previous calculation. Plugging these into the point-slope form gives \( y - (-5) = -1(x - 2) \), which simplifies to \( y + 5 = -x + 2 \). This equation now represents the unique line through the given point with the specified slope, and we can easily convert it to other forms as needed for different applications.

Linear Equations

Finally, the grand concept tying everything together is the linear equation itself. These equations form lines when graphed on a coordinate plane and have the general form \( Ax + By = C \), where A, B, and C are constants. This is known as the standard form of a linear equation.

To get from the point-slope form to the standard form, we perform algebraic manipulations to eliminate fractions and get the x and y terms on one side of the equation. In our example, rearranging \( y + 5 = -x + 2 \) gives us the standard form \( x + y = -3 \).

The ability to fluently convert between different forms of linear equations (like slope-intercept, point-slope, and standard form) is essential for tackling various algebra problems. It also has practical applications in fields such as engineering, physics, and economics, where understanding the relationships between variables is critical.
To improve comprehension, one helpful exercise is to take different pairs of points and practice writing their corresponding linear equations in all three forms. These core concepts form the bedrock of much of the algebraic understanding necessary for advanced mathematics studies.