Problem 40 - Algebra 1

Question

Write an equation in standard form of the line that passes through the given point and has the given slope. $$(5,-8), m=10$$

Step-by-Step Solution

Answer

The equation of the line in standard form is \(10x - y - 58 = 0\).

Step 1: Write the equation in slope-intercept form

The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. We know the slope (\(m = 10\)) and a point on the line ((5, -8)). Plugging these values into the equation, we have \(-8 = 10 * 5 + b\), solving this for \(b\) will give the y-intercept.

Step 2: Solve for b

Solving the equation \(-8 = 50 + b\) for \(b\) gives \(b = -8 - 50 = -58\). Thus, the equation in slope-intercept form is \(y = 10x - 58\).

Step 3: Rewrite in standard form

The standard form of a linear equation is \(Ax + By = C\), where \(A, B,\) and \(C\) are integers, \(A\) and \(B\) are not both zero, and \(A\) is nonnegative. To rewrite the equation from slope-intercept form to standard form, we can move the terms around to get \(10x - y - 58 = 0\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line's equation is one of the most commonly used formats because it quickly provides crucial information about the line. This form is expressed as \(y = mx + b\), where:

• \(m\) represents the slope of the line, which describes its steepness and direction. A positive slope means the line rises from left to right, while a negative slope means it falls.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

To find an equation in slope-intercept form, you need to know the line's slope and a point on the line. This information allows you to solve for the y-intercept. For example, with a slope \(m = 10\) and a point \(5, -8\), we can substitute these values into the formula to find \(b\):

• Substitute into \(y = mx + b\) → \(-8 = 10 \times 5 + b\)
• Solve for \(b\) → \(b = -58\)

This calculation gives us the complete equation in slope-intercept form: \(y = 10x - 58\). Knowing this form helps visualize the line's graph and understand how it behaves across different values of \(x\).
Understanding the slope-intercept form is crucial, especially when transitioning to other forms of linear equations.

Standard Form

The standard form of a linear equation offers another way to write the equation of a line. It is expressed as \(Ax + By = C\), where:

• \(A\), \(B\), and \(C\) are integers.
• \(A\) and \(B\) cannot both be zero, and \(A\) should ideally be a non-negative integer.

This form is particularly useful for various applications, such as solving systems of equations or when integer coefficients are preferred over fractions. To convert an equation from slope-intercept to standard form, you can rearrange it to move all terms to one side:

• Start from \(y = 10x - 58\).
• Rearrange to bring all terms to one side → \(10x - y = 58\).
• Optionally, write as \(10x - y - 58 = 0\) to match the standard form's typical arrangement.

Converting to standard form helps mainly in cases where two linear equations need to be handled simultaneously, and it offers a cleaner, integer-based approach. Understanding both this form and the slope-intercept form equips you to express linear equations in ways that best fit different mathematical contexts.

Linear Equation

Linear equations are fundamental mathematical expressions capturing the concept of a straight line. Regardless of the form they are presented in, whether slope-intercept or standard, their essence remains centered around constant coefficients and a single variable whose degree is one. Key properties include:

• They graph as straight lines, illustrating a constant rate of change.
• The general formula is \(Ax + By = C\), but can be seen as \(y = mx + b\) for convenience.
• Known for simplicity, featuring no exponents, square roots, or products of variables.

Linear equations offer a starting point in understanding more complex mathematical topics. They help in visualizing relationships between variables and interpreting real-world phenomena as mathematical models. Given a slope \(m\) and a point, learning how to transition between different forms like slope-intercept and standard gives clarity on how equations represent lines on a graph. Being comfortable with these transformations is useful for solving practical problems, whether for plotting the course of a journey or analyzing trends in data over time. Linear equations form the backbone of algebra, and mastering them unlocks the door to advanced mathematical exploration.