Problem 38 - Algebra 1

Question

Write an equation in standard form of the line that passes through the given point and has the given slope. $$(10,6), m=7$$

Step-by-Step Solution

Answer

The equation in standard form of the line is \(7x - y = 64\).

Step 1: Write the Point-Slope Form of the Line

The point-slope form of a line is \(y - y_1 = m(x - x_1)\) where \(m\) is the slope and \((x_1, y_1)\) are the coordinates of the given point. By substituting \(m = 7\), \(x_1 = 10\) and \(y_1 = 6\) into the equation, we get: \(y - 6 = 7(x - 10)\).

Step 2: Distribute the Slope on the Right Hand Side

We multiply 7 by each term inside the parenthesis to get: \(y - 6 = 7x - 70\).

Step 3: Convert to Standard Form

Standard form of a line is \(Ax + By = C\), where A, B and C are integers. We can convert the equation to that form by rearranging it: \(7x - y = 64\).

Key Concepts

Point-Slope Form

Understanding the point-slope form of a line is essential for writing equations that represent a line in coordinate geometry.

The point-slope formula \(y - y_1 = m(x - x_1)\) offers a direct method to write an equation of a line when you know the slope \(m\) and a point \( (x_1, y_1) \) on the line. Here's the formula at a glance:

• \((y_1)\) is the y-coordinate of the known point.
• \((x_1)\) is the x-coordinate of the same point.
• \((m)\) represents the slope of the line.

\((y - y_1\)) and \((x - x_1\)) show the distance from any point \((x, y)\) on the line to the known point. The equation intuitively indicates how far up or down you need to go \( (y - y_1)\) and how far left or right \( (x - x_1)\) to stay on the line, as guided by the slope, \(m\).

For the given problem, the line goes through the point \( (10, 6) \) and has a slope of 7. Substituting these values into the point-slope form, we get the equation of the line as it relates to its slope and a specific point on the line. The ease of using the point-slope form lies in its straightforward substitution process, making it a favorite for quickly drafting the framework of a linear equation.

Slope of a Line

The slope of a line is a measure that indicates how steep the line is and its direction.

Mathematically, the slope \(m\) is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between two distinct points on a line: \[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \].

• If the slope is positive, the line rises from left to right.
• If the slope is negative, it falls from left to right.
• If the slope is zero, the line is horizontal.
• If the slope is undefined, the line is vertical.

In our exercise, the given slope is 7, which means for every one unit the line moves horizontally to the right (positive x-direction), it rises by 7 units in the vertical direction (positive y-direction). This characteristic of the slope defines the inclination of the line and is key to building the equation that represents such a line. The procedure of using slope values can also be applied to find the equation of parallel or perpendicular lines, as they will have slopes that are either equal or negative reciprocals, respectively.

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are used to describe lines in the coordinate plane.

These equations can be represented in various forms, including:

• Standard form: \(Ax + By = C\), with \(A, B\), and \(C\) as integers, and \(A\) should be positive.
• Slope-intercept form: \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
• Point-slope form: \(y - y_1 = m(x - x_1)\) as discussed earlier.

The standard form is particularity useful for analyzing equations quickly, identifying intercepts and is required in certain academic contexts. Converting to standard form, as done in Step 3 of our exercise, involves moving all terms involving variables to one side of the equation and the constant to the other side, while also keeping the leading coefficient positive. In our provided exercise, rearranging the point-slope form equation resulted in the standard form \(7x - y = 64\), which expresses the relationship between \(x\) and \(y\) in an equivalent, yet different, manner than the point-slope form, showcasing the versatility of linear equations in mathematics.