Problem 23 - 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. \((-1,4), m=-3\)

Step-by-Step Solution

Answer

The line that passes through the point (-1,4) and has slope -3 is represented by the equation \(3x + y = 1\) in standard form.

Step 1: Start with the Slope-Intercept Form of an Equation

The slope-intercept form of a linear equation is defined as \(y = mx + b\), where m is the slope, and b is the y-intercept. We are given the slope \(m = -3\), and the point which the line crosses, \((-1,4)\). Therefore, we can substitute these values into the equation \(y = -3x + b\). Now, the main task is to find the y-intercept \(b\).

Step 2: Find the Y-Intercept

Insert the coordinates of the given point into the equation from step 1 in order to find the y-intercept. We have \((-1,4)\), so when \(x = -1, y = 4\). Therefore, \(4 = -3(-1) + b\). Solving for b gives \(b = 4 - 3 = 1\). Hence, the equation in slope-intercept form is \(y = -3x + 1\).

Step 3: Transform into Standard Form

The standard form of an equation is \(Ax + By = C\), where A must be positive and A, B, and C are integers with no common factors other than 1. We have \(y = -3x + 1\) from step 2. Rearranging the terms, we get \(3x + y = 1\). So, the equation of the line in standard form that passes through the point \((-1,4)\) and has a slope of -3 is \(3x + y = 1\).

Key Concepts

slope-intercept form

The slope-intercept form is a way to write the equation of a line so you can easily identify its slope and y-intercept. In this form, the equation of a line is written as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) stands for the y-intercept, which is where the line crosses the y-axis.
To understand the slope \(m\), think of it as "rise over run" – how much y increases (or decreases) as you move along the x-axis. For example, if the slope is 3, the line rises 3 units upward for every 1 unit it goes to the right.
The slope-intercept form is especially useful because it gives immediate visual information about a line plotted on a graph. By looking at it, you can predict the rate of change of y in relation to x and know exactly where it will intersect the y-axis. This form allows you to easily plug in any given point and slope to find out the line's equation.

standard form of a line

The standard form of an equation is another method to express a linear equation, typically written as \(Ax + By = C\). In this format, \(A\), \(B\), and \(C\) must be integers, and \(A\) should be positive. One of the advantages of using the standard form is that it simplifies the process of finding x- and y-intercepts directly when graphing.
Unlike the slope-intercept form, standard form highlights the relation between x and y without giving an immediate indication of the slope or y-intercept. However, it is versatile in solving systems of equations and can deal with vertical and horizontal lines more conveniently.
By rearranging the slope-intercept form, such as \(y = -3x + 1\), into standard form, we turn it into \(3x + y = 1\). This transformation helps in situations where you are required to solve equations algebraically, like finding points of intersection.

finding y-intercept

The y-intercept is the point where a line crosses the y-axis, meaning it occurs when \(x = 0\). In the equation \(y = mx + b\), the y-intercept is given by \(b\), the constant term. Finding the y-intercept is straightforward if you know the equation of the line since it's where x equals zero.
When you're given a point and the slope, you can find the y-intercept by plugging these values into the slope-intercept equation and solving for \(b\). For instance, using the point \((-1,4)\) with a slope of \(-3\), you substitute into the equation \(4 = -3(-1) + b\). Solving this gives \(b = 1\), meaning the line crosses the y-axis at \((0,1)\).
Knowing the y-intercept helps in sketching the line and understanding its position in relation to the axis. It's a crucial part of graphing linear equations and makes the overall approach much clearer. Finding it solidifies the full equation and assists in better visualizing and interpreting the line on a graph.