Problem 29 - Algebra A Combined Function

Question

Find an equation of each line with the given slope that passes through the given point. Write the equation in the form $A x+B y=C. $$ m=-\frac{1}{2} ; \quad(-3,0) $$

Step-by-Step Solution

Answer

The equation of the line is \( x + 2y = -3 \).

Step 1: Formula Selection

To find the equation of the line, we start by using the point-slope form of a linear equation, which is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is the given point.

Step 2: Substitute the Values

Substitute the given slope \( m = -\frac{1}{2} \) and the point \((-3, 0)\) into the point-slope equation: \( y - 0 = -\frac{1}{2}(x + 3) \).

Step 3: Simplify the Equation

Simplify the equation: \( y = -\frac{1}{2}(x + 3) \), which becomes \( y = -\frac{1}{2}x - \frac{3}{2} \).

Step 4: Clear Fractions

Multiply every term by 2 to eliminate fractions: \( 2y = -x - 3 \).

Step 5: Rearrange into Standard Form

Bring all terms to one side to achieve the form \( Ax + By = C \): \( x + 2y = -3 \).

Step 6: Final Equation

The equation of the line in standard form is \( x + 2y = -3 \).

Key Concepts

Point-Slope Form

The point-slope form is a way to express the equation of a line. It is very useful when you know the slope of the line and one point on the line. The formula is given by:

• \( y - y_1 = m(x - x_1) \)

Here, \( m \) represents the slope, while \((x_1, y_1)\) is the coordinates of the known point.
This form allows you to quickly plug in the values you have and find the equation of the line. Since it directly incorporates the slope and a point, it can be easier to use than other forms when these two pieces of information are given.
For example, if you have a slope \( m = -\frac{1}{2} \) and a point \((-3, 0)\), you can substitute these directly into the formula to get an initial equation.

Slope-Intercept Form

Another common way to write the equation of a line is the slope-intercept form. This format is called slope-intercept because it directly shows the slope of the line and the y-intercept (where the line crosses the y-axis). The equation looks like this:

• \( y = mx + b \)

In this formula, \( m \) is again the slope, and \( b \) represents the y-intercept.
This form is very handy for graphing because it quickly tells you two things - how steep the line is (slope) and where it starts on the y-axis (y-intercept).
If we simplify the point-slope equation \( y = -\frac{1}{2}x - \frac{3}{2} \), we have it in slope-intercept form: the slope \( m \) is \(-\frac{1}{2} \) and the y-intercept \( b \) is \(-\frac{3}{2} \).

Standard Form (Ax + By = C)

Standard form is a way to write the equation of a line using integers. The general equation is:

• \( Ax + By = C \)

In this form, \( A \), \( B \), and \( C \) are integers, and \( A \) should be a positive number.
It is particularly useful in various analytical and practical applications, such as solving systems of equations.
To rewrite an equation from another form into standard form, you often need to rearrange the terms and eliminate fractions.
For example, starting with \( y = -\frac{1}{2}x - \frac{3}{2} \), by multiplying everything by 2 to remove fractions and rearranging, we finally get \( x + 2y = -3 \) as the standard form.

Slope

The slope of a line is a measure of its steepness. In mathematical terms, it is often denoted by \( m \) and calculated based on the change in y over the change in x (often called "rise over run").
The slope formula is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.
A positive slope means the line rises as you move from left to right, while a negative slope means the line falls. A zero slope indicates a horizontal line, and an undefined slope corresponds to a vertical line.
In our case, the given slope is \(-\frac{1}{2}\), meaning the line falls by \(\frac{1}{2}\) unit vertically for every unit it moves horizontally.