Problem 75 - Algebra and Trigonometry Real Mathematics, Real People

Question

Equation of a Line in Standard Write the standard form of the equation of the line that has the specified characteristics. \(m=4,\) passes through (-1,2)

Step-by-Step Solution

Answer

The standard form of the equation of a line that has the slope \(4\) and passes through the point \(-1, 2\) is \(-4x + y = 6\)

Step 1: Write the equation using point-slope form

The point-slope form of a linear equation is \(y - y_1 = m(x - x_1)\) where \(m\) is the slope and \((x_1, y_1)\) is the point given. So, substituting the given slope \(m=4\), and the point given (-1,2) into the point-slope form, we get \(y - 2 = 4(x + 1)\)

Step 2: Simplify the equation to the slope-intercept form

Let’s distribute 4 in the equation we just found and move the -2 to the other side to find the y-intercept form. Thus, it becomes \(y = 4x +4 +2\) or \(y = 4x + 6\)

Step 3: Convert the equation to the standard form

The standard form of a linear equation is \(Ax + By = C\), where A, B, C are integers (positive or negative whole numbers) and A and B are not both zero. Starting from the slope-intercept form \(y = 4x + 6\), we need to move the x-term to the left side. By subtracting \(4x\) from both sides we then get the standard form: \(-4x + y = 6\).

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a format where you can quickly write the equation of a line if you know its slope and one of its points. This form can be very handy in situations where you need to write an equation on the fly. The formula is given by:

• \(y - y_1 = m(x - x_1)\)

Here, \(m\) represents the slope of the line, and \((x_1, y_1)\) is a specific point on the line.
For example, if you have a line with a slope of \(4\) that goes through the point \((-1, 2)\), you substitute these into the formula to get:

• \(y - 2 = 4(x + 1)\)

This equation tells us a lot about the line:

• The line is steep because of the slope of \(4\).
• The line passes through \((-1, 2)\).

Once you have the point-slope form, you can use it to find other forms of the equation, such as slope-intercept form or standard form.

Slope-Intercept Form

The slope-intercept form of a line is especially useful for understanding and graphing linear equations. This form is represented by the formula:

• \(y = mx + b\)

In this formula, \(m\) is the slope of the line again, and \(b\) is the y-intercept, which tells you where the line crosses the y-axis.
Converting the point-slope form to slope-intercept form involves distributing any constants, then simplifying. For the line we’ve been working with, the process goes like this:

• Start with \(y - 2 = 4(x + 1)\).
• Distribute the \(4\) inside the parentheses: \(y - 2 = 4x + 4\).
• Move the constant \(-2\) to the other side: \(y = 4x + 4 + 2\).
• Simplify to get: \(y = 4x + 6\).

With this equation, it’s easy to see that:

• The line rises quickly since \(m = 4\).
• The y-intercept is \(6\), so the line crosses the y-axis at point \((0, 6)\).

Standard Form of a Linear Equation

The standard form of a linear equation is useful for analyzing relationships between two variables and is expressed as:

• \(Ax + By = C\)

Where \(A\), \(B\), and \(C\) are integers, and \(A\) and \(B\) shouldn't be both zero. This form is very clean and can be beneficial in problem-solving, particularly when dealing with systems of equations.
To convert from the slope-intercept form into the standard form for our equation, follow these steps:

• Start with the slope-intercept form: \(y = 4x + 6\).
• Subtract \(4x\) from both sides to get \(-4x + y = 6\).

Now, our equation \(-4x + y = 6\) is in standard form, but it can often be written with positive \(A\) values. If desired, multiply each term by \(-1\):

• That change gives us: \(4x - y = -6\).

Using the standard form:

• It becomes easier to identify both variables \(x\) and \(y\).
• This form is particularly useful in algebraic equation solving scenarios.