Problem 11 - Algebra 1

Question

Write an equation in standard form of the line that passes through the given point and has the given slope. $$(1,-2), m=5$$

Step-by-Step Solution

Answer

The equation of the line in standard form is \(5x - y = 7\).

Step 1: Plug in given data into point-slope form

The first step is to substitute the given point \((1,-2)\) and slope \(5\) into the point-slope form \(y - y1 = m(x - x1)\), which gives \(y - (-2) = 5(x - 1)\).

Step 2: Simplify equation

The next step is to simplify the equation. First, deal with the minus before the two, which results in \(y + 2 = 5x - 5\). Then, get the constants to the right side of the equation to get \(5x - y = 7\).

Step 3: Check the equation is in standard form

Lastly, check that your equation, \(5x - y = 7\), is in standard form, \(Ax + By = C\), where \(A > 0\). In this case, it is since \(A = 5,\) \(B = -1\) and \(C = 7\) are all integers, and \(A\) is positive.

Key Concepts

Standard Form of a Line

The standard form of a line is an essential concept in algebra that describes a straight line on a Cartesian coordinate system. When we talk about writing linear equations, the standard form is one of the ways to express these equations. The hallmark of standard form is its structure, written as
\(Ax + By = C\)
where \(A\), \(B\), and \(C\) are integers, and \(A\) should be a non-negative integer. One of the main reasons to use standard form is that it clearly demonstrates the intercepts on the axes, as ℎ4Responses to the X and Y variables are isolatable by algebraic manipulation. Here are some additional points to bear in mind:

• If \(A\) is positive, it's in the preferred standard form.
• If either \(A\) or \(B\) is zero, the line will be either horizontal or vertical respectively.
• The integer \(C\) represents the constant, and it shows the point where the line crosses the Y-axis when X is zero (known also as the Y-intercept).

Let's take the exercise example:\(5x - y = 7\). The coefficients fit perfectly into the standard form requirements: \(A = 5\), \(B = -1\), and \(C = 7\), with \(A\) being positive. It's straightforward, concise, and ready for further analysis or graphing.

Point-Slope Form

Point-slope form is invaluable when you know the slope of a line and a single point it passes through. This form is less common in final answers but is instrumental in finding the equation of a line during the initial stages of problem-solving. The point-slope formula is given as
\(y - y_1 = m(x - x_1)\)
where \((x_1, y_1)\) is a point on the line, and is the slope of the line. Here's why this form is so useful:

• It directly incorporates the characteristics of the line given a point and a slope.
• It provides a straightforward way to begin writing the equation before converting it to standard or slope-intercept form.

The exercise provided demonstrates the use of the point-slope form. Given the point \((1, -2)\) and the slope \(5\), it's plugged into the formula like so: \(y - (-2) = 5(x - 1)\). After simplification, the equation transforms into a more familiar linear equation—preparing it for conversion to standard form.

Linear Equations

A cornerstone of algebra, linear equations form the basis for understanding how quantities are related in a straight-line manner. Linear denotes a relationship that graphs to a straight line, and in mathematics, these relationships are described by linear equations. They come in many forms, such as the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, and the already discussed standard and point-slope forms.
Linear equations have the following characteristics:

• They are polynomial equations of the first degree (their highest power of the variable is one).
• They graph as straight lines on the Cartesian coordinate system.
• They can model numerous real-world situations, from simple things like determining the cost of goods to complex scientific relationships.

Understanding how to manipulate linear equations and interchange between their different forms allows for flexibility in solving and applying these equations in various contexts, much like adjusting the exercise's equation from point-slope to standard form.