Problem 26 - 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. $$ m=-2 ; \quad(-11,-12) $$

Step-by-Step Solution

Answer

The equation is \(2x + y = -34\).

Step 1: Identify the Slope-Intercept Form

The point is given as \((-11, -12)\) and the slope \(m\) is \(-2\). The equation of a line in slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, \(m = -2\).

Step 2: Substitute the Point into the Equation

Using the point \((-11, -12)\), substitute \(x = -11\) and \(y = -12\) in the equation, which gives us:\[-12 = -2(-11) + b\]

Step 3: Solve for the Y-Intercept (b)

Simplify the expression from Step 2:\[-12 = 22 + b\]Subtract 22 from both sides to solve for \(b\):\[b = -34\]

Step 4: Write the Equation in Slope-Intercept Form

Using the slope \(m = -2\) and \(b = -34\), the equation of the line in slope-intercept form is:\[y = -2x - 34\]

Step 5: Convert to Standard Form

To convert the equation into standard form \(Ax + By = C\), rearrange the slope-intercept equation:\[y = -2x - 34\]Add \(2x\) to both sides:\[2x + y = -34\]This is now in the standard form \(Ax + By = C\) where \(A = 2\), \(B = 1\), and \(C = -34\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of an equation of a line is a very popular way to represent linear equations, especially when we want to quickly identify the slope and the y-intercept of a line. The general format is given by the formula \(y = mx + b\), where:

• \(m\) denotes the slope of the line
• \(b\) is the y-intercept, the point where the line crosses the y-axis

This form is extremely useful because it allows us to understand how steep a line is (through the slope) and where it starts on the y-axis. For example, in the exercise, with a point \((-11, -12)\) and a slope of \(-2\), we substitute the slope and point into our formula to find the y-intercept. Once calculated, we have the equation of the line in slope-intercept form as \(y = -2x - 34\). This form directly shows us that the line has a slope of \(-2\) and intersects the y-axis at \(-34\).
Recognizing and using the slope-intercept form is the first critical step in finding the equation of a line efficiently.

Point-Slope Form

Point-slope form is another valuable representation of the equation of a line, especially useful when you know a specific point on the line and the slope. It is expressed as:

• \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the known point and \(m\) is the slope.

This form is handy because it directly incorporates the coordinates of a known point along with the slope, making it simple to work with. For instance, in our problem, given the point \((-11, -12)\) and slope \(-2\), we'd start with:\[y + 12 = -2(x + 11)\].Expanding the equation will yield a more familiar format, but this form helps in visualizing how to "anchor" a line on a graph using a single point and allows further manipulation into other forms if needed.
The point-slope form is especially effective in creating an equation immediately from specific data points.

Standard Form

Standard form of a linear equation is organized as \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are integers, and typically, \(A\) and \(B\) are non-negative with \(A\) not equal to zero. This form is particularly useful in many mathematical settings, such as solving systems of linear equations. It sets the stage nicely when combining multiple equations or finding intersections.To convert from slope-intercept form like \(y = -2x - 34\) (found in the previous step), we rearrange the terms:
1. Add \(2x\) on both sides to isolate terms with variables on the same side: \(2x + y = -34\).
This equation is now perfectly aligned with the standard form, easily readable as \(2x + y = -34\), where \(A = 2\), \(B = 1\), and \(C = -34\).
Standard form is ideally suited for analytical purposes, such as plotting a line's interaction with other geometrical elements.