Problem 56 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given point and has the given slope. $$ (0,5), m=-1 $$

Step-by-Step Solution

Answer

The point-slope form of the equation of the line that passes through the point (0,5) and has the slope -1 is \(y = -x + 5\).

Step 1: Identify the given values

Identify and label the given point and slope. In this case, the point \((x_1, y_1)\) is (0,5) and the slope \(m\) is -1.

Step 2: Substitute values into the formula

Substitute the values into the point-slope form equation \(y - y_1 = m(x - x_1)\). This gives \(y - 5 = -1(x - 0)\). This simplifies to \(y - 5 = -x\).

Step 3: Rearrange the equation

Rearrange the equation to the standard point-slope form equation which results in \(y = -x + 5\).

Key Concepts

Equation of a Line

An equation of a line is a mathematical statement that expresses the relationship between the x and y coordinates on the Cartesian plane. A line equation allows us to understand where the line lies in the plane, its angle of inclination, and its behavior as x and y values change.
The most commonly used forms for line equations are:

• Slope-Intercept Form: \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.
• Point-Slope Form: \(y - y_1 = m(x - x_1)\) which uses a known point \((x_1, y_1)\) and the slope \(m\) of the line.
• Standard Form: \(Ax + By = C\) which provides a neat way to represent linear equations, especially in systems of equations.

Among these, the point-slope form is particularly useful when you have a specific point on the line and the line's slope. It directly aligns with these two factors, offering a straightforward way to formulate the line's equation.

Slope

The slope of a line is a measure that describes both the direction and the steepness of the line. Represented by the letter \(m\), the slope is calculated as the "rise" over the "run." This refers to how much the y-coordinate of a point on the line changes as the x-coordinate changes.

• A positive slope means the line goes upward as it moves from left to right.
• A negative slope means the line goes downward as it moves from left to right.
• A slope of zero indicates a horizontal line.
• Undefined slope corresponds to a vertical line.

The formula to find the slope \(m\) given two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\].
In our exercise, the slope is given as \(-1\), indicating the line falls at a 45-degree angle as it moves to the right.

Coordinates

Coordinates are a set of values that show an exact position on a plane. In a two-dimensional space, they are written as \((x, y)\), where \(x\) corresponds to the horizontal axis and \(y\) corresponds to the vertical axis.
Understanding coordinates is vital for:

• Locating Points: Coordinates allow us to locate any point on the Cartesian plane. For example, the point \((0, 5)\) is exactly 5 units up on the y-axis.
• Graphing Lines: Marking points and using them to draw or construct lines on a graph.
• Analyzing Shapes: Understanding how different points are situated helps analyze geometric shapes on the plane.

In the problem we are considering, the origin point \((0, 5)\) represents a known location for our line, critical for forming our equation using the point-slope form.

Linear Equations

Linear equations are polynomials of the first degree, meaning the highest power of the variable is 1. These equations form straight lines when graphed on a coordinate plane. Recognizing and working with linear equations is a fundamental math skill, assisting not just in math but also in real-life modeling.

• Characteristics: A linear equation in two variables (x and y) looks like \(ax + by = c\), with graphical representation as a straight line.
• Solutions: The solutions to a linear equation are ordered pairs, \((x, y)\), which satisfy the equation.
• Applications: Used in calculating speeds, finance (like interest and profit problems), and all regular situations needing straight-line representation.

Transforming our point-slope form equation into a slope-intercept form allows us to directly see its linear function form. The conversion of \(y - 5 = -x\) to \(y = -x + 5\) illustrates how this line behaves on the graph.