Problem 27 - Algebra 1: Concepts and Skills

Question

Write in standard form an equation of the line that passes through the given point and has the given slope \((2,9), m=\frac{2}{5}\)

Step-by-Step Solution

Answer

The standard form of the equation of the line that runs through the point \((2,9)\) and has a slope of \(\frac{2}{5}\) is \(2x - 5y = -41\).

Step 1: Apply Point-Slope Formula

First, place the given slope and coordinates into the point-slope formula. In this case, \(x_1 = 2\), \(y_1 = 9\), and \(m = \frac{2}{5}\). So the formula becomes \(y - 9 = \frac{2}{5}(x - 2)\).

Step 2: Simplify the Equation

Next, distribute the \(\frac{2}{5}\) on the right side to get \(y - 9 = \frac{2}{5}x - \frac{4}{5}\). After that, add 9 to both sides, which will give \(y = \frac{2}{5}x + \frac{41}{5}\).

Step 3: Convert To Standard Form

The standard form of a linear equation is \(Ax + By = C\), with \(A, B \neq 0\), and \(A > 0\). Multiply the entire equation by 5, the denominator of the fractions, to eliminate the fractions and to get into standard form. After doing so, the equation becomes \(2x - 5y = -41\).

Key Concepts

Point-Slope Formula

The point-slope formula is a handy tool for writing the equation of a line when you know the slope and a point on the line. The point-slope formula is written as:

• \( y - y_1 = m(x - x_1) \)

Here, \( m \) represents the slope of the line, and \( (x_1, y_1) \) are the coordinates of the given point through which the line passes. This form is especially useful because it directly incorporates the slope and point, thereby making it intuitive for crafting linear equations.
The point-slope form can quickly transform into other forms of linear equations, such as slope-intercept or standard form. Thus, it's a versatile starting point when tackling problems involving linear equations.

Linear Equations

Linear equations describe straight lines on a graph. A linear equation is typically expressed in one of several forms, including point-slope, slope-intercept, and standard form. Each form provides different insights and is useful in various situations.
Linear equations like \( Ax + By = C \) (standard form) do not involve exponents beyond 1, and they graph as straight lines. These equations can often be solved graphically or algebraically to find the intersection points, intercepts, or solutions to the system of equations.
Understanding linear equations is crucial for solving real-world problems where relationships between quantities are constant.

Slope-Intercept Form

The slope-intercept form of a linear equation is popular for its simplicity and ease of use. It is presented as:

• \( y = mx + b \)

Here, \( m \) represents the slope of the line, which is the ratio of the vertical change to the horizontal change between any two points on the line. The \( b \) value is the y-intercept, the point where the line crosses the y-axis.
This form makes it very straightforward to sketch a graph, as you can easily identify the slope and the y-intercept just by glancing at the equation. Moreover, converting between the various forms of a linear equation often involves getting to or from this form because of its simplicity and informativeness.