Problem 8 - Elementary Algebra

Question

For Problems 1-12, find the equation of the line that contains the given point and has the given slope. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. $$ (-3,9), m=0 $$

Step-by-Step Solution

Answer

The equation of the line is \(0x + y = 9\).

Step 1: Understanding the Problem

We need to find the equation of a line that passes through the point (-3, 9) and has a slope \(m = 0\). Since the slope is zero, the line is horizontal.

Step 2: Using Point-Slope Form

A horizontal line has a constant \(y\)-value for all \(x\)-values. Since the slope \(m = 0\), the horizontal line equation is \(y = k\), where \(k\) is the \(y\)-coordinate of the given point. For the point (-3, 9), \(y = 9\).

Step 3: Converting to Required Form

To express the equation \(y = 9\) in the form \(Ax + By = C\), rearrange it to \(0x + 1y = 9\). This can be written as \(0x + y = 9\), which shows \(A = 0\), \(B = 1\), and \(C = 9\).

Step 4: Final Equation

Thus, the equation of the line in the standard form \(Ax + By = C\) is \(0x + y = 9\).

Key Concepts

Point-Slope Form

The point-slope form is a way of writing the equation of a line when we know a point on that line and its slope. It's very handy in algebra for describing linear relationships quickly. The general formula is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope.
This form helps us see how the slope affects the line's rise over run, literally showing the difference between two points, which relates to slope, in the equation.
When applying the point-slope form, it becomes straightforward to substitute the known values – the point and the slope – directly into the formula to find the line's equation. In our exercise, because the slope equals 0, the situation simplifies significantly.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It's often described as "rise over run," meaning how much the line goes up or down (rise) for each unit it moves to the right (run). It's usually represented by the letter \( m \).

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right.
• If the slope is zero, the line is flat and horizontal.

In our case, the slope is 0, indicating a horizontal line. This means no matter how much the \( x \)-value changes, the \( y \)-value stays constant. Understanding the slope is crucial, as it tells us the behavior of the line on a graph.

Horizontal Line Equation

A horizontal line is unique because its slope is always 0. This quality simplifies its equation significantly. Remember, a slope of 0 means there's no change in the \( y \)-values regardless of changes in the \( x \)-values.
For any point \( (x_1, y_1) \) on a horizontal line, the equation becomes \( y = y_1 \). It indicates that every point on the line has the same \( y \)-coordinate. For our example, the point (-3, 9) suggests that the line equation is \( y = 9 \). This matches our knowledge that horizontal lines are constant in their \( y \)-coordinate.
To express this in the standard form \( Ax + By = C \), simply recognize that \( y = 9 \) can be seen as \( 0x + 1y = 9 \), giving these integer values: \( A = 0 \), \( B = 1 \), and \( C = 9 \). This conversion helps when different forms are required to solve algebra problems.